
Glass, 
Book. 



COPYRIGHT DEPOSIT 



"The dexterous hand and the thoughtful mind find their strength in union alone" 

ROGERS' '^ 

Drawing and Design 

An Educational Treatise 

RELATING TO 

LINEAR DRAWING; MACHINE DESIGN; WORKING DRAWINGS; TRANSMISSION 
METHODS; STEAM, ELECTRICAL AND METAL WORKING MACHINES AND PARTS; 
LATHES; BOILER AND PARTS; INSTRUMENTS AND THEIR USE; TABLES, Etc. 




THEO. AUDEL & CO., 72 FIFTH AVENUE, NEW YORK 

PUBLISHERS 



-COPYRIGHTED BY 



THEi. AUllL 



NEW YORK, 1913 



ALL RIGHTS RESERVED 



^0. (M3 

_©CI,A85128 5 



■35 3 




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iiniiiiiiiiiiiiiitt 



3 

I 



" One peculiar feature of the draftsman's opportunity is that it takes hold 
of all the mechanical occupations, and of one almost as much as of the other. 
It is not in the least monopolized by the machinist, and it is not the necessity 
of his shop more than of the others. The pattern maker certainly has quite as 
much to do with working drawings, and why not also the molder, the black- 
smith, the boiler maker, the carpenter, the coppersmith and all the rest? It 
will be to the immense advantage of the workers in any of these lines, and to 
the young man a most presumptive means of advancement, to be not only able 
to read drawings, but to make them." — American Machinist. 



PREFACE 

In a report to the Bridge Commissioners, as to the progress being made in the construction of the 
steel cables designed to support the immense weight of the (N. Y.) East River Suspension Bridge, 
Chas. G. Roebling, C. E., used these impressive words, quoted, as printed in The Sun : 

" Further, Mr. Roebling said the work of placing four cables nineteen inches in diameter across the river, was one that 
REQUIRED A CERTAIN DELIBERATION. No ERROR OP ANY KIND MUST BE MADE. Although all the men that could be utilized 
in the work have been employed, yet the progress made appears to be slow. Laymen might, from this, infer that the work is 
lagging, but the Commissioners should know that this was not the case. The work will proceed, Mr. Roebling says, and be 
finished to the perfect satisfaction of the Commissioners." 

These emphasized words have been frequently in the mind of the author of this work, relating to 
drawing and design. During the long months required by its preparation the greatest of care has 
been taken to avoid error of any sort, and the utmost deliberation has been given to the careful 
presentation of each subject. 

This has been called "the age of illustration ;" the truth of the saying is evident on every side 
from the daily illustrated newspaper to the blue print in the hand of the iron worker. In illustrations 
of whatever nature we come back to the L. B. & T. elements — length, breadth and thickness — and to 
linear drawing as the foundation of all drawing whether industrial or artistic ; for linear drawing has for 
its object the accurate delineation of surfaces and the construction of figures obtained by the studied 
combination of lines. We must come back to first principles in all knowledge, as the ball comes 
back to the hand of the skillful thrower, so that on the next attempt it may be projected still further 
upward. 



PREFACE. 



The ability to draw is like an added sense whose value could be somewhat determined, if those 
engineers and others who are skilled in the art, would name the sum of money for which they would 
part with its knowledge — for — 

" A chance sketch — the jotted memoranda of a contemplative brain often forms the nucleus of. a 
splendid invention. An idea thus preserved at the moment of its birth may become of incalculable 
value when rescued from the desultory train of fancy and treated as the sober offspring of reason." 
This quotation is from the one who wrote also the noble sentence — " Thou hast not lost an hour whereof 
there is a record ; a written thought at midnight will redeem the livelong day." 

From its inception to its closing page the main idea of the author has been to instruct, to impart 
knowledge of drawing and design with special reference to a considerable degree of method and 
completeness ; his aim has been to educate, or to draw out, and develop harmoniously the mental 
powers — to train to a certain result the various processes described and to nurture an abiding interest 
in the student's mind of a noble and ancient art. "First, the blade, then the ears, then the ripened 
corn appears " expresses what has been the attempted order of instruction. 

The power to draw is akin to that, and, to the engineer and mechanic, second only to the power to 
read ; one needs not only to read the printed page but also to read a blueprint or a rightly drawn and 
porportioned sketch ; there should be many thousand good draughtsmen scattered about and around 
before there is one professor or regular instructor m the art ; for to the average man drawing should be 
looked upon as a help in his daily avocation rather than as a staff to lean upon for life-long employment. 



PREFACE. 

There is a current saying, "one never sees an old draughtsman." This is more true in the United 
States — the home of Opportunity — than in older countries ; its meaning is that the position in the 
draughting room is but a stage in the development of the Engineer, the Superintendent, the Manager 
in engineering works. 

A good knowledge of draughting is a round on the ladder of preferment ; a second round is a fair 
working knowledge of tJie mathematics and theory of mechanism, for the foundation of all accurate 
attainment in drawing and design are laid in these two fundamental sciences. It may be well always 
to remember that 

" Education does not consist merely in storing the head with materials ; that makes a lumber room of it ; but in learning how 
to turn those materials into useful products ; that makes a factory of it ; and no man is educated unless his brain is a factory, 
^vith storeroom, machinery and material complete." 

To this may be added that the helpful value of a teacher or instructor cannot be overestimated ; man 
was not created to do his appointed work alone, he needs all assistance and aid possible — to help and be 
helped — is the universal law of progress, and especially is this true in the first beginnings in the art of 
drawing ; afterwards the student may be supposed to have acquired a real interest in his stimulating 
and useful endeavors. 

It is an odd thing that the preface, which is always understood as something going before, is often 
written last, hence these few long paragraphs are prepared to close the long and rather pleasant task 
of the author of the book ere it is delivered iyt toto to the Printer, Binder and to the management of 
its most excellent and reliable Publishers for its introduction to those for whom it is designed. 

With these views and to further such ends this book has been prepared, and with such aims more 
or less successfully attained, the volume is now committed to the kind favor of its patrons. 



TABLE OF CONTENTS 



PAGES 

Plan of the Work 

Abbreviations and Conventional Signs - 25-26 

Useful Terms and Definitions - - - 27-40 

Drawing Board, T-Square and Triangles - 41-51 

Lettering 52-64 

Shade Lines 65-77 

Section Lining - 77-8 1 

Geometrical Drawing - - - - 83-110 

Isometric Projection 111-120 

Cabinet Projection 121-127 

Orthographic Projection - - . . 128-161 

Development of Surfaces - - - 162-179 

Working Drawings " 181-187 

Tints and Colors 188 

Tracing and Blue Printing - , - - 189-195 

Reading of Working Drawings - - 196-198 

Machine Design - - - - . - 199-228 



PAGES 

Physics and Mechanics - - - - 212-228 
Material Used in Machine Construction 214-215 
Screws, Bolts and Nuts - - - - 228-242 
Rivets and Riveted Joints - - - 243-251 

Power Transmission 253-255 

Shafts and Bearings 256-266 

Belts and Pulleys, ----- 266-277 

Gear Wheels 278-304 

Metal Working Machines - - - 305-332 
Dies and Presses - - . - - . 308-314 
Drilling AND Milling Machines - - 31S-319 

The Lathe 320-332 

Engines and Boilers - - - - 333-389 

Electrical Machines - - - - - 391-407 
Drawing Instruments .... 408-426 

Logarithms 435-460 

Tables and Index - - - - . - 461-486 



THE SCOPE AND PLAN OF THE WORK, 



The special mission of this book can almost be gathered from its title page and the preface. It 
is intended to furnish gradually developed lessons in linear drawing applied to the various branches of 
the mechanic arts. 

The work is comprised within some twelve divisions or general subjects ; the first of which consists 
of Abbreviations and Conventional Signs, Useful Terms and Definitions with illustrations. 

The second section relates to the Drawing Board, T-square and Triangles and their use, lettering, 
shade and section lining, etc. 

The third division is devoted to Geometrical Drawing ; the subject is preceded by many valuable 
definitions, axioms and examples of postulates and followed by many illustrations, largely based upon 
the problems solved by Euclid more than twenty-two centuries ago. 

The fourth division relates to the Development of Surfaces and Isometric, Cabinet and Orthographic 
projections. The fifth section relates to Working Drawings embracing Tracing, Blue Printing, 
Dimensioning, Reading of Drawings, etc. 

The foregoing portions comprise "Part One" of the work and relate almost exclusively to 
Drawing and Definitions. "Part Two" is devoted to Machine Design, Transmission Methods, 
Metal Working Machinery, Engines and Boilers, Electrical Machines, etc., which embrace the sub- 
divisions six to ten. 

Each one of these sections is preceded by explanatory matter, and accompanied by illustrations of 
the different machines, with working directions for proportioning and designing. 

"Part Three," in addition to Drawing Instruments and their U^^e and the Index, contains 
tables, of the utmost value, for use in connection with the preceding sections, especially so, as the 
basis of the work is planned to be largely mathematical. 



ROGERS' DRAWING AND DESIGN. 



The making of a book of any considerable scope and value is either — as in olden times — the life 
work of a single author, or as, at the present, the combined efforts of several individuals, whose 
united efforts produce it in a much shorter time, and it must be hoped, in greater perfection. 

Although no more than a year has elapsed from the opening subject of "Abbreviations and 
Conventional Signs" to the closing reference — Index — pages, in no sense should the work be 
considered as being hasty or superficial, for the outcome of the combined efforts of those named below, 
is worthy of praise for having produced a thoroughly scientific and helpful book. 

First of all, to Mr. John Weichsel, M. E., instructor in drawing and design in one of the foremost 
technical institutes of New York City, is due the credit of furnishing most of the drawings and 
diagrams used throughout the work, with the text accompanying each ; the book itself is the highest 
testimonial to the admirable and thoroughly technical character of Prof. Weichsel's work. 

Mr. Henry E. Raabe, M. E., has been the technical editor throughout the period of the prepara- 
tion of the text and the arrangement of the illustrations in their appropriate places. Many of the 
drawings, explanatory notes, and "cuts" are also his own production. 

Messrs. Sutherland & Graham, Engravers, and George Byron Kirkham, Artist, are entitled to 
thanks for many designs and illustrations, as well as for professional advice and suggestions in several 
details of the "lay-out" of the volume. Mr. P. Hetto, of the U. S. Navy, an accomplished draughts- 
man and scholar, has read the "proof" of each separate page with critical care, and to him should be 
accorded praise for the almost perfect freedom from errors of any kind which marks the completed 
volume. 

Mr. H. Harrison, of the L. Middleditch Press, has used his wide experience in the typographical 
arrangement of the work; in this he has been aided by Mr. Henry J. Harms in overseeing the final 
issue and printing of the book ; the excellence of their work is evident on every printed page. 

It may be added that the kind and experienced editor-in-chief has combined and added to, and in 
some cases, taken from, the "matter" submitted by the foregoing named persons and others and the 
result of the whole, is now ofTered with confidence to the patronage of the Mechanical World, by 

The Publishers. 



ABBREVIATIONS AND CONVENTIONAL SIGNS* 



In order to simplify the language or expression of arithmetical and geometrical opera- 
tions the following conventional signs are used : 

The sign -f- signifies plus or more and is placed between two or more terms to indicate 
addition. Example: 4 -|- 3, is \ plus 3, that is, 4 added to 3, or 7. 

The sign — signifies minus or less and indicates subtraction. 

Example : 4 — 3, is 4 minus, that is, 3 taken from 4, or i. 

The sign x signifies multiplied by and placed between two terms, indicates multiplication. 
Example : 5 X 3, is 5 multiplied by 3, or 15. 

When quantities are expressed by letters, the sign may be suppressed, thus we write 
indifferently, a x b or ab. 

The sign : or (as it is more commonly used) -4- signifies divided by, and, placed between 
two quantities, indicates division. 

Example : 12 : 4, or 12 -7- 4 or — is 12 divided by 4. 

The sign = signifies equals or equal to, and is placed between two expressions to indi- 
cate their equality. Example : 6 + 2 = 8, meaning that 6 plus 2 is equal to 8. 



ABBREVIATIONS AND CONVENTIONAL SIGNS. 



The union of these signs : :: : indicates geometrical proportion. 

Example : 2 : 3 : : 4 : 6, meaning that 2 is to 3 as 4 is to 6. 



The sign /^Z indicates the extraction of a root ; as, 

-y/ 9 = 3, meaning that the square root of 9 is equal to 3. 

The interposition of a numeral between the opening of this sign, V, indicates the degree 
of the root. Thus: ^^27 = 3. expresses that the cube root of 27 is equal to 3. 

The signs < and > indicate respectively, smaller than and greater than. 

Example : 3^4 = 3 smaller than 4 and, reciprocally, 4 > 3 = 4 

greater than 3. 

Fig. signifies figure ; and pi., plate. 



USEFUL TERMS AND DEFINITIONS. 

Lines, Angles, Surfaces and Solids constitute the different kinds of quantity called geometrical magnitudes. 



LINES AND ANGLES. 

A surface is that which has extension in length 
and breadth only. 

A solid is that which has extension in length, 
breadth and thickness. 

An an^le is the difference in the direction of 
two lines proceeding from the same point. 

A point is said to have position without magni- 
tude ; it is generally represented to the eye by a 
small dot 

A line is considered as length without breadth 
or thickness ; it denotes the direction between two 
points. Lines are principally of three kinds : (i) 
right lines, (2) curved lines, (3) mixed lines. 



FiG.l. 



A right line, or as it is more commonly called, 
a straight line is the shortest line that can be 
drawn between two given points, as above in Fig. i. 

A curved line is one of which no portion, how- 
ever small, is straight ; it is therefore longer than a 
straight line connecting the same points ; a straight 



line is often called simply a line, and a curved line 
a curve, a regular curved line, as Fig. 2, is a portion 




Fig. 2. 



of the circumference of a circle, the degree of curva- 
ture being- the same throuo-hout its entire leng-th ; an 
irreo^tilar curved line has not the same degfree of 
curvature throughout, but varies at different points. 

A -waved line, shown in Fig. 3, may be either 



regular or 



FiQ. 3. 

irregular; the illustrat.on shows the 
former, the inflections on either side of the dotted 
line being equal. 

Note. — There are other lines used in common drawing-room defini- 
tions, viz.: Broken, etc. 

Bi oketi — One composed of different successive straight lines. Center 
— A line used to indicate the center of an object. Conshuction — A 
working line used to obtain required lines. DoUed — A line composed 
of short dashes. Dash — A line composed of long dashes. Dot and 
Dash — A line composed of dots and dashes alternating. Dimension — 
A line upon which a dimension is placed. Full — An unbroken line, 
usually representing a visible edge. Shade — A line about twice as 
wide as the ordinary full line. 



27 



28 



ROGERS' DRAWING AND DESIGN. 



Mixed lines are composed of straight and curved 
lines, as Fig. 4. 



FiQ. 4. 



Parallel lines are those which have no inclina- 
tion to each other — Figs. 5 and 6 being everywhere 
equidistant ; consequently they never meet though 
produced to infinity. If the parallel lines shown in 
Fig. 6 were produced they would form two circles 
having a common center. 



Fig. 5. 




Fig. 6. 

Horizontal lines are lines parallel with the hori- 
zon, as in Fig. 7. 

Vertical lines are often called plumb lines as 
they are parallel to a plumb line suspended freely 
in a still atmosphere. A horizontal line in a draw- 
ing is shown by a line drawn from left to right 



across the paper ; a vertical line in a drawing is 
represented by a line drawn up and down the paper 
or at right angles to a horizontal line, as in Fig. 7. 



HORIZONTAL 




Fig. 7. 



Inclined or oblique lines occupy an inter- 
mediate between -horizontal and vertical lines as 
shown in Fig. 7 ; two lines which converge towards 
each other and which, if produced, would meet or 
intersect are said to incline towards each other. 




ROGERS' DRAWING AND DESIGN. 



29 



Perpendicular lines. Lines are perpendicular 
to each other when the angles on either side of the 
point of meeting are equal. 

Vertical and horizontal lines are always perpen- 
dicular to each other, but perpendicular lines are 
not always vertical and horizontal ; they may be 
at any inclination to the horizon, provided that the 
angles on either side of the point of intersection 
are equal, as X Y and Z in Fig, 8. 



/ 



/ 









s. 


«•■ 


-9 


1 

1 


i 


i 


f 


\ 


/ 


■p 


r 


-7 


<>•• 


■^*-> 


e-^ 



Fig. a. 



Fig. 10. 



Angles. Two straight lines drawn from the 
same point, diverging from each other form an 
angle, as shown in Fig. 9 : the angle is the differ- 
ence in the direction of two lines proceeding from 
the same point. 

Note. — Mechanics' squares, if true, are always right-angled. 



A right angle is formed when two straight 
lines intersect so that all angles formed are equal, 
as shown in Fig. 10. 

An obtuse angle is 

greater than a right angle, \,^o^'"^'^f^ 
Fig. II. 






Fig. U. 



An actite angle is 

smaller than a right angle, 
Fig. 12. 

Obtuse and acute angles are also called 
oblique angles ; and lines which are neither parallel 
nor perpendicular to each other 
are called oblique lines. 

The vertex or apex of an 

angle is the point in which the 
including lines meet. 

An angle is commonly desig- 
nated by three letters and the letter designating 
the point of divergence is always placed in the 
middle. 

The magnitude of an angle 
does not depend upon the 
length of the sides but upon 
their divergence from each 
other. 




Fig. 12. 




Fig. 13. 



30 



ROGERS' DRAWING AND DESIGN. 



STRAIGHT SIDED FIGURES. 

A surface is a magnitude that has length and 
breadth without thickness; as a plane surface, or, 
the imaginary envelope of a body. 

A plane is a surface such that if any two of its 
points be joined by a straight line, such line will be 
wholly in the surface. Every surface which is not 
a plane, or composed of plane surfaces, is a curved 
surface. 




Fig. 14. 



Fig. 15. 



Fig. 18. 



A plane figure is a portion of a plane terminated 
on all sides by lines either straight f)r curved. 

A rectilinear figure is a surface bounded by 
straight lines. 

Polygon is the general name applied to all 
rectilinear figures but is commonly applied to those 
having more than four sides. A regular polygon is 
one in which the sides are equal. 



A triangle is shown in Fig. 13. When sur- 
faces are bounded by three straight lines they are 
called triangles. 




Fig. ir. 



An equilateral triangle has all its sides of 
equal length, and all its angles equal, as appears 
in the illustration. Fig. 13. 



Fig. 18. 



An isosceles triangle has two of its sides and 
two of its angles equal, as illustrated in Fig. 14. 



ROGERS- DRAWING AND DESIGN. 



31 



A right-angled triangle has one right angle, 
Fig. 15 : the side opposite the right angle is called 
the hypothenuse ; the other sides are called respec- 
tively the base and perpendictilar. 

The altitude of a triangle is the length of a 
perpendicular let fall from its vertex to its base. 




iTia. n. 



Fig. 20. 




Fio. 21. 



Fio. 22. 



A quadrilateral is a figure bounded by four 
straight lines. If the opposite sides of a quad- 
rilateral are paralleled it is* called a parallelogram. 

Note. — The superficial conlents of a triangle may be obtained b}' 
multiplying the altitude by one half the base. 



A parallelogram, in which the four sides are equal, 
and form right angles with each other, is called a 
square. Fig. 16. 

There are three kinds of quadrilaterals : The 
trapezium, the trapezoid, and the parallelogram — 
as above. 

The trapezium has no two of its sides parallel, 
Fig. 17. 

The trapezoid has only two of its sides parallel, 
Fig. 18. 

There are four varieties of parallelograms : The 
rhomboid, the rhombus, the rectangle and the 
square. 

The square is an equilateral rectangle, Fig. 
16. 

A rhombus is a parallelogram as shown in Fig. 
19, one in which the four sides are equal, but none 
of the angles are right angles. 

A rectangle is a parallelogram which has its op- 
posite sides parallel, and all its angles right angles, 
Fig. 20. 

A rhomboid is a parallelogram in which the 
adjacent sides are unequal, and none of the angles 
are right angles, Fig. 21. 



32 



ROGERS' DRAWING AND DESIGN. 



A diagonal is a straight line, which joins two 
opposite angles of a polygon, Figs. 17, 22. 

A pentagon is a polygon bounded by five 
straight lines, Fig. 23. If the sides and angles 
formed by them are equal the figure is called a reo-- 
tilar pentagon. Fig. 24. 




Fig. 23. 



Fig. 24. 



Fig. 2.-). 





Fig. 26. 



Fig. 27. 



Fig. 2/i. 



A hexagon is a polygon bounded by six straight 
lines. Fig. 25 illustrates a regular hexagon. 



A heptagon is a polygon bounded by seven 
straight lines. Fig. 26 illustrates a regular hep- 



4agon. 



An octagon is a a polygon bounded by eight 
straight lines. In Fig. 27 is shown a regular 
octagon. 

A decagon is a polygon bounded by ten straight 
lines. Fig. 28 illustrates a regular decagon. 




Fig. 29. 

A dodecagon is a polygon bounded by twelve 
straight lines. In Fig. 29 is shown a regular dodec- 
agon. 

Note. — Polygons of more than eight sides are rarely used in me- 
chanical drawing. Their most frequent application occurs in laying 
out of the hubs of large sectional wheels. 



ROGERS" DRAWING AND DESIGN. 



33 



A convex surface is one that when viewed from 
without is curved outward by rising or swelHng into 
a rounded form, Fig. 30. 

CONVEX. 



Fig. 30- 

A double convex stirface is regularly protuberant 
or bulging on both sides. 




Fig. 31. 



Concave means hollow or curved inward ; said 
of an interior of an arched surface or curved line in 
opposition to convex, Fig. 31. 



CIRCLES AND THEIR PROPERTIES. 
A circle is a plane figure bounded by one uni- 
formly curved line, all of the points in which are at 
the same distance from a certain point within, called 
the center ; the circumference of a circle is the 
curved line that bounds it ; the diameter of a circle 
is a line passing through its center, and terminating 
at both ends in the circumference ; the raditis of a 
circle is a line extending from its center to any 
point in the circumference ; it is one-half of the 



diameter ; all the diameters of a circle are equal, as 
are also all the radii ; an arc of a circle is any portion 
of the circumference ; the fixed point about which 
the circle is drawn is called the center of the circle ; 
any straight line, drawn within the circle, connect- 




ing any two points in the circumference without 
passing through the center, is called a chord. 

A semicircle is the half of a circle and is 
bounded by half the circumference and a diameter ; 
a segment of a circle is any part of its surface cut off 
by a straight line ; a sector of a circle is a space 
included between two radii and the arc they inter- 
sect. See Fig. 32. 

Note. — Radius is derived from the Latin word ray, meaning a di- 
vergent line, the plural in Latin is radii ; the English word for the 
plural term is radiuses. 



34 



ROGERS' DRAWING AND DESIGN. 



A quadrant is a sector equal to one-fourth of 
the circle ; the two radii bounding a quadrant are at 
right angles. 

A tangent to a circle or other curve is a straight 
line which touches it at only one point. Every 
tangent to a circle is perpendicular to the radius 
drawn to the point of tangency. 

A degree. The circumference of a circle is sup- 
posed to be divided into 360 equal parts called degrees 
and marked (°). Each degree is divided into 60 
minutes, or 60'; and for the sake of still further mi- 
nuteness of measurement, each minute is divided into 
60 seconds, or 60". In a whole circle there are, 
therefore, 360X60X60^1,296,000 seconds. The 
annexed diagram, Fig. 32, exemplifies the relative 
positions of the 

Sine, Tangent, 

Co-Sine, and Co-Tangent 
of an angle ; the co- in co-sine and co-tangent is 
simply an abbreviation of the word, complement. 
The circumferences of all circles contain the same 
number of degrees, but the greater the radius the 
greater is the absolute measures of a degree, and 
every circumference is the measure of precisely the 
same angle. 

Thus if the circle be large or small, the number 



of the division is always the same, a degree being 
equal to -jiirth part of a circle ; the semicircle is 
equal to 180° and the quadrant to 90°. 

The sine of an arc is a straight line drawn from 
one extremity perpendicular to a radius drawn to 
the other extremity of the arc, Fig. .32 ; the co-sine 
of an arc is the sine of the complement of that arc, 
as shown in the same figure. 

The tangent of an arc is a line which touches 
the arc at one extremity and is terminated by a line 
passing from the center of the circle through the 
other extremity of the arc, Fig. 32 ; the co-tangent 
of an arc is the tangent of the complement. 

For the sake of brevity, these technical terms are 
contracted thus : for sine, we write sin.; for co-sine, 
we write cos.; for tangent, we write tan., etc. 




Fig. 33. 



Concentric circles are those which are de- 
scribed about the same center, Fig. 33. 



ROGERS' DRAWING AND DESIGN. 



35 



Mccentric circles are those which are described 
about different centers, Fig. 34. 





Fig. 34. Fig. Sj. 

Eccentric circles are two or more circles whose 
centers lay within the circumference of one or 
more of these circles, but do not form a common 
center about which they could all be described. 
Figs. 34, 35- 36. 





Fig. 36. Fio. 37. 

The centers of eccentric circles may also lay out- 
side of each other's circumference, as in Fig. 37, 
or the center of one circle may lay outside of the 



other's circumference, while the former circle may 
be either wholly or partly within the circumference 
of the latter, as in Figs. 38 and 39. 





Fig. 38. 



Fig. 39. 



In another instance the center of one circle may 
lay on the circumference of the other, as in Fig. 40. 




Fig. •40. 



The distance between the centers of eccentric 
circles is called the radius of eccentricity. 



36 



ROGERS' DRAWING AND DESIGN. 



If two circles lay in a position as indicated in 
Fig. 41, they are not regarded as eccentric circles, 
but are treated as two independent figures. 




Fig. 41. 



A Parabola is a curve, described by a point, 
moving so, that its distances from a straight line, 
and a fixed point are always equal, Fig. 42 ; the 




Fig. 42. 



Straight line is called the Directrix, and the fixed 
point is called the Focus of the parabola ; a straight 
line drawn at right angles to the directrix, and pass- 
ing through the focus, is called the Axis. 




A Hyperbola is a curve from any point of which, 
if two straight lines be drawn to two fixed points, 
their difference shall always be the same. See 
Fig- 43- 



^ 



Fig. 44. 



An Mllipse is a curve, described by a point, mov- 
ing so, that the sum of its distances from two fixed 
points is always constant ; the two fixed points are 
called.- the Foci, Fig. 44. 



ROGERS' DRAWING AND DESIGN. 



37 



SOLIDS. 

A solid has the parts constituting its substance 
so compact or firmly adhering as to resist the im- 
pression or penetration of other bodies ; it has a 
fixed form, is hard, firm and unHke a fluid or liquid ; 
it is not hollow, hence sometimes heavy. 

A conic section is a curved line formed by the 
intersection of a cone and a plane. 

Intersection of solids is a term used to de- 
scribe the condition of solids virhich are so joined 
and fitted to each other as to appear as though one 
passes through the other 

By the envelope of a solid is meant the surface 
which encases or surrounds it. 

A prism is a solid body whose ends or bases are 
equal and parallel plane figures, and whose sidt s 
are parallelograms. The shape of a prism is always 
expressed by the form of its bases. 

A triangular prism is a prism with the trian- 
gular bases, as shown in Fig. 45. 

A quadrangular prism is a prism with quad- 
rangular bases. Fig. 46. 

A pentagonal prism is a prism with pen- 
tagonal bases, Fig. 47. 



A hexagonal prism is a prism with hexagonal 
bases. Fig. 48. 






Fig. 45. Fiu. 4U. Fio. 4T. 

A cube'is a quadrangular prism whose b;ises and 
sides ar ' all equal and form perfect squares, Fig. 49. 





Fic;. 4S. 



Fli:. 41), 



Flti. .ill. 



A cylinder is a solid, bounded by two equal 
circular surfaces or bases, and one continuous curved 
surface, Fig. 50. All cross sections of a cylinder 
are equal to the bases. 



38 



ROGERS' DRAWING AND DESIGN. 



A cone is a solid bounded by a circular base, 
and one curved surface, extending from the circular 
base to a point opposite it, Fig. 51. 




Fig. 51. PiQ. 52. Fig. 53. 

Aright cone. If a perpendicular, droppedfrom 
the apex of the cone to its base, meets the center 
of the base circle, the cone is called a right cone, 
Fig. 52. The perpendicular in this case is called 
the Axis of the cone. 

An oblique cone. If the perpendicular falls 
alongside the center of the base circle, or entirely 
outside of its circumference, the cone is called an 
oblique cone. Fig. 53. 

A truncated cone. A cone, cut off in the man- 
ner shown in Fig. 54, is called a truncated cone, 
If an oblique cone is cut in the above manner, it is 
called an oblique truncated cone. Fig. 55. 



If a cone is cut by a plane, parallel to the outline 
of its surface, vertically opposite the center line of 
the cutting plane, as shown in Fig. 56, the outline 
of the section is 2. parabola. Fig. 57. 




Fig. 54. 




Fig. M. 





Fio. 57. 



ROGERS" DRAWING AND DESIGN. 



39 



If the cutting plane forms a smaller angle than 
the parabola, with the outline of the side on which 
A 





Fig. 59. 



it is cut, as shown in Fig. 58, the section is a hyper- 





FiG. 60. Fig. 61. 

bola. Fig. 59. If the cutting plane forms a greater 
angle than the parabola, with the surface, so that 



the cone is cut in the manner shown in Fig. 60, the 
s?ction is an ellipse, Fig. 61. 

t^ pyramid is a solid, whose base is a polygon, 
and whose sides are formed by triangles. 

The point in which all the lines of the triangular 
sides meet, is called the apex of the pyramid. 

APEX 




Fig. 62. Fig. 63. Fig. 64. Fig. 65. 

Pyramids are classified as triangular, quadrangu- 
lar, pentagonal, hexagonal, etc., pyramids, depend- 
ing upon the shape of their base, Figs. 62, 63, 64. 

If the base of a pyramid forms a regular polygon, 
and a perpendicular dropped from the apex, to the 
base, passes through the center of the base, it is 
called a right pyramid. Fig. 65. 

The altitude of a pyramid or a cone, is the per- 
pendicular distance from the apex to the base. Figs. 
66, 67. The altitude of a prism or a cylinder is the per- 
pendicular distance between the bases, Figs. 68, 69. 



40 



ROGERS' DRAWING AND DESIGN. 



A truncated pyramid is the part remaining, 
after the apex is cut away, Figs. 54 and 70. A 
truncated cone or pyramid is also called the 
frustrtim of the cone or pyramid. 




Fig. tj6. 



Fig. 70. 



Fig. 67 




Fig. 68. 



Fig. 




Fig. 71. 



A sphere is a solid, bounded by a uniformly 
curved surface, any point of which is equidistant 
from the center, Fig. 71. 



A polyhedron is a solid, bounded by polygons. 
There are five regular polyhedrons — as follows : 

A tetrahedron is a solid, bounded by four equi- 
lateral triangles, Fig. 72. 

A hexahedron is a solid, bounded by six 
squares ; the common name for this solid is cube, 
Fig. 49. 

An octahedron is a solid, bounded by eight 
equilateral triangles. Fig. y^- 






\ 





Fig. 72. 



Fig. 73. 



A dodecahedron is a solid, bounded by twelve 
regular pentagons. 

An icosahedron is a solid, bounded by twenty 
equilateral triangles. 



DRAWING BOARD, T-SQUARE AND TRIANGLES. 



The problems explained in the following para- 
graphs are but a small part of the great number of 
problems that may be executed with the aid of 
the tee-square and triangles. 




Fio. 74. 



In fact, all drawings, embracing straight lines only, 
may be drawn with the aid of the above instru- 
ments, provided the nature of the drawing does not 
call for greater accuracy or for lines other than 
straight ones. In the latter case, the mathematical 



instruments described hereafter need to be em- 
ployed. 

The paper on which it is intended to make a 
drawing, is generally fastened, by means of thumb- 
tacks, to a specially made board called a drawing 
board, Fig. 74. 

The drawing board should be made about 2 
inches longer and 2 inches wider than the paper. It 
should be made of well-seasoned, straight-grained 
pine, free from all knots ; the grain should run 
lengthwise of the board. 

The edges of the board should be square to each 
other and perfectly smooth in order to provide a 
good working edge for the head of the tee-square 
to slide against. 

A pair of hard-wood cleats is screwed to the back 
of the board. The board should be about three- 
quarter inch in thickness. The cleats, fitted at 
the back of the board, at right angles to its long- 
est side, may be about two inches wide and one 
inch thick. Such cleats will keep the board from 
warping through changes of temperature and moist- 
ure. 



4i 



42 



ROGERS' DRAWING AND DESIGN. 



Fig. ' 



Fig. 76. 



All lines parallel to the longer edges 
of the board are called horizontal lines. 
For drawing such lines an instrument is 
used, called a tee-square, Fig. 75. A tee- 
square consists of two parts, the head 
and the blade, which should be square to 
each other. 

The blade should be as long as the 
drawing board. It should be made of 
well-seasoned, fine-grained hard wood, 
and as light as its proper use will permit. 
The head may be made of any kind of 
well-seasoned wood. The blade should 
be laid on the face of the head and there 
fastened to it with four or five screws. 
The tee-square should be used with its 
head held firmly against the left hand 
edge of the board. Any number of hori- 
zontal lines may be drawn by sliding the 
tee-square up or down, Fig. 76. 

Another kind of a tee-square is shown 
in Fig. 'J 'J. The blade of this tee-square 
is fastened to the head by means of a 
square-necked bolt and a fly-nut. The 
blade may be so adjusted as to form any 
desired angle with the head. This tee- 
square is called the adjustable tee-square. 



ROGERS' DRAWING AND DESIGN. 



43 




Fig. 77. 




Fig. 78. 



After setting the blade at the desired angle 
to the head, we can draw any number of 
parallel lines at that angle, by sliding the 
tee-square up or down. Fig. 78. 

For drawing lines other than horizontal 
ones, set squares or triangles are used. 
They are made in various styles, some being 
cut out of a single piece of wood, while 
others are framed together of three or more 
pieces. 

Two triangles will be required for ordi- 
nary purposes. One should have one angle 
of 90 degrees, that is a square angle or a 
right angle, and two angles of 45 degrees 
each, that is equal to one-half of a right 





Fig. 79. 



Fig. 80. 



44 



ROGERS' DRAWING AND DESIGN. 



angle ; the two short sides of the triangle are of 
equal length, Fig. 79. 

The other triangle should contain one angle of 
90 degrees, one angle of 30 degrees (that is equal 




Fig. 81. 



to one-third of a right angle) and one angle of 60 
degrees (that is equal to two-thirds of a right 
angle.) In this triangle the shortest side is equal 
to just one half the longest side, Fig. 80. 



The first triangle is called the 45-degree triangle, 

the second, the 30-degree triangle or the 60 degree 

triangle. These triangles may be made of wood, 

hard rubber, or celluloid, of which materials it is 

also preferable to make the tee-square 

for many reasons. Triangles made of 

straight-grained well-seasoned hard wood 

will be found most satisfactory. 

By placing the tee-square in position on 
the drawing board, with its head against 
the left-hand edge of the board, and plac- 
ing either triangle with its short side to the 
edge of the tee-square, we may draw lines 
parallel to the short side of the drawing 
board, which we will call vertical lines. Fig. 
81. Any number of vertical lines may be 
drawn by sliding the triangle in this posi- 
tion along the edge of the tee-square. 

The manner in which the head of a tee- 
square is united to the blade determines 
its adaptability or otherwise to the use 
made of it ; in some the head of a tee- 
square is rectangular in section, and the 
blade mortised into it ; in others the blade is dove- 
tailed and let into the head of a tee-square for the 
whole of its thickness ; the method spoken of on 
page 42 is, however, the most approved. 



ROGERS' DRAWING AND DESIGN. 



45 



Keeping the tee-square in position and placing 
against its blade one edge of the 45-degree angle, 
we may draw a line making an angle of 45 degrees 
with a horizontal line, Fig. 82. Such a line is called 



o 
o o 




Fig. 82. 



a 45-degree line, or we may write it 45° line, the 
small circle at the top, placed after the number 
meaning degree. 

By placing one edge of the 30-degree angle 
against the blade of the tee-square, held in position 
on the board, a line making an angle of 30 degrees 



with a horizontal line, or simply a 30-degree line, 
Fig. 83, may be drawn ; in a similar manner a 
60-degree line may be drawn with the 60-degree 
angle of the triangle. 

By combining the two triangles as in 
Figs. 84, 85, a 15-degree line and a 75- 
degree line may be drawn. 

We may draw a line or lines parallel 
to any given line in our drawing, by the 
use of the two triangles in the following 
manner : 

Place one of the triangles with one of 
its edges exactly on the given line ; place 
the longest edge of the second triangle 
against the longer one of the two remain- 
ing edges of the first triangle; then hold 
the second triangle in place and slide 
upon it the first triangle. Fig. 86. 

Using the triangles in a similar manner 
we may draw a line or lines at a right 
angle to anv triven line, thus — 

Place one edge of either triangle exactly on the 
given line ; place the longest edge of the second 
triangle exactly to the longest edge of the first 
triangle. Hold the second triangle in place and 
turn the first triangle so that one edge will form a 



O 



46 



ROGERS' DRAWING AND DESIGN. 



right angle with the given line, as in Fig. 87. By 
placing one edge of the right angle of either tri- 
angle on the given line, as the first operation, the 
first trianele will not have to be turned. 




6 



By sliding the first triangle upon the second one, 
any number of lines may be drawn which will be at 
right angles to the given line. 

With a knowledge of the preceding rules a great 







ROGERS' DRAWING AND DESIGN. 



47 



variety of figures may be drawn. In the following 
we will show how a square, an equilateral triangle, 
a hexagon and an octagon may be drawn by these 
simple means. 




GIVEN LINZ 




Flu. 86. 




Fio. 87. 




ROGERS' DRAWING AND DESIGN. 




Let it be required to draw a four-sided plane 
figure, all sides of which are equal and all its angles 
right angles. Such a figure is called a square. 

If the sides of the required square should be 
parallel to the edges of the drawing board, we then 
draw a horizontal line by means of the tee-square. 
Fig. 88. On this line we mark two points, the dis- 
tance between them being equal to the side of the 
required square. By means of either triangle draw 
vertical lines through the two points on the first 



ROGERS' DRAWING AND DESIGN. 



49 



line ; make one of the vertical lines equal to the 
first line, that is equal to a side of the square, by 
drawing a 45-degree line from the foot of one of 
the vertical lines, to meet theother vertical line, and 
move the tee-square to the point of intersection of 
these two lines, where a horizontal line is drawn, 
meeting the other vertical line and forming the con- 
cluding side of the required square. 

If a square is to be drawn on any given line, 
which is neither horizontal nor vertical (such lines 
are called oblique lines) we will proceed as follows : 
Fig. 89, on the given line mark two points, the dis- 
tance between them being equal to a side of the re- 
quired square. Through these two points draw 
lines at rig-ht angfles to the first line. Make one of 
these sides equal to a side of the required square 
and draw through the end of it a line parallel to 
the first line. This line will form the concluding 
line of the square. 

Let it be required to draw a square, when the 
length of its diagonal only is given, Fig. 90. 

Place the longest edge of the 45-degree triangle 
exactly on the given diagonal. Place the tee-square 
with an edge of its blade against one of the short 
sides of the triangle. By sliding the triangle upon 



the tee-square held in place, draw two lines through 
the ends of the given diagonal with the other short 
edcre of the triano-le. These will form two sides of 
the square. Bring the triangle back to its first posi- 




FlG. 90. 



tion upon the diagonal, hold it in place and remove 
the tee-square, placing it now against the other short 
edge of the triangle. Sliding the triangle upon the 
tee-square, draw the two remaining sides of the 
square, as before. 



50 



ROGERS' DRAWING AND DESIGN. 



To draw an equilateral triangle upon a given 
line. The angles in an equilateral triangle are 60- 
degree angles, Fig. 91. 




Fig. 91. 



Place the edge of the blade of the tee-square ex- 
actly on the given line. Place one edge of 60-de- 
gree angle of the 60-degree triangle to the edge of 
the tee-square, and draw lines making an angle of 
60 degrees with the tee-square, through both ends 
of the given line. These lines, with the given line, 
will form the required triangle. 

To draw a regular hexagfon : 

Draw a line, AB, Fig. 92 ; set off from any 
point, O, on the line AB, two distances, AO 
and OB, each equal to a side of the required hex- 
agon. Through the points A, O and B draw six 




parallel lines making angles of 60 degrees with the 
line AB ; three lines, AE, DOC and FB in one 
direction, and the other three, CB, EF and AD in 
the other direction. Join the points E and C and 
D and F. A E C B F D is the hexagon required. 

To draw a hexagon on a given line. Let AB be 
the given line. Fig. 93. Draw the lines AC and 
BE at an angle of 60 degrees to the given line, in 
one direction, and the lines AF and BD, at the 
same angle, in the other direction. At the points 
A and B on the given line, draw two lines at right 
angles to this line, these lines cutting the lines, EB 
and AF, at the points E and F. Join E and F 



ROGERS' DRAWING AND DESIGN. 



51 



and through the points E and F draw the Hnes EC 
and FD at 6o-degree angles to the given Hne, EC 
cutting the line AC at C and FD cutting the line 
BD at D. Then A C E F D B is the required 



hexagon. 




To draw an octagon on a given line : 

Let AB be the given line, Fig. 94. At the points 

A and B draw lines at angles of 45 degrees to the 

criven line, AC, in one direction, and BH in the 

other direction. Make AC and BH each equal to 



the line AB. Through C draw the line CF parallel to 
BH and through H the line EH parallel to AC. 
Draw lines through C and through H at right angles 
to the given line ; the line CD cutting the line BD 




Fig. 94. 



at D and the line HG cutting the line AG at G. 
Through D draw the line DE parallel to CF and 
cutting EH at E ; then draw through G the line 
GF parallel to EH and cutting CF at F. Join EF 
and ACDEFGHBis the required octagon. 



52 



ROGERS' DRAWING AND DESIGN. 



ALPHA BETA ANTiqVA 2rasE5?;Siso 








f 



Fig. 95. 



ROGERS' DRAWING AND DESIGN. 



53 



LETTERING. 

Let it be said that lettering is intended to convey 
to the mind of the observer a simple but attractive 
impression of what the drawing is to express. 

When the information necessary to the reading 
of a drawing cannot be expressed by lines and 
scale dimensions, it must be indicated in the form 
of printed explanations, remarks, etc., as explained 
and illustrated in the following pages. 

Whole volumes have been published upon this 
most fascinating subject. When writing was the 
universal mode of expression, that is, before the in- 
vention of printing — the art of lettering was one of 
the fine arts. Many manuscripts are now extant 
whose titles are made upon vellum in inks of gold, 
scarlet, blue and other gaudy colors ; these have 
added vastly to the value of the books and aided in 
their preservation through the long centuries. The 
illustration upon the opposite page is given as a 
specimen of one of these ancient {^'antique") 
alphabets. 

To do good lettering is not an easy task, and 
unless the student is already experienced he should 
devote much time to practicing the art, working 
slowly and bearing in mind that much time is 
required to make well-finished letters. 

Lettering of various styles are in use, some quite 



simple and others difficult, especially in cases where 
an ornamental heading is required, but it must be 
remembered that a drawing is primarily made to 
convey an idea, and not for an ornament. 

The character and size of the letters on all work- 
ing drawings should be in harmony with the draw- 
ing on which they appear. It is desirable to have 
all lettering on a drawing made in the same style, 
only differing in size or finish of details. 

Capital letters should always be sketched in 
pencil, especially by the beginner, and inked in 
afterwards ; the lettering used on mechanical draw- 
ings is usually of the simplest character, the letters 
being composed of heavy and light strokes only ; 
for headings, titles of large drawings, where com- 
paratively large lettering is required, it will be most 
appropriate to use large letters. 

The title should be conspicuous, but not too much 
so ; sub-titles should be made smaller than the 
main-title. 

The " Scale " and general remarks placed in the 
margin of the drawing or near the title should 
come next in size. All explanations and remarks 
on the views should not be larger than one-eighth 
inch. 

The examples of lettering given as illustrations 
are briefly explained on page 63. 



54 



ROGERS' DRAWING AND DESIGN. 



n " 


"IS ''■ .S " 


:::5 ::::: 


::ii:::2::i5 : :::: : 


— hv- 


"]a:_:_!.__ 




- >- ^-1^ 


— :[?3i:- 


::3i::::i: 


" II ^ 


11^ :: III - ~ 


-I 3t-iC 


"^-t 




X :: 


zz-l-Az 


±S.i:i:: 


i:::^::±:: 


:::ii :s^:f3::::::::::: 



^ 




TT Un^ 



N U1 I I !SI 



IS 



m 



m 



a 






^~^ 



s 







s 







» 




5 



tzt 



Fig. 90. 



E 



;^ 



iii 



IS 



I 



I 



s 



FW 



? 



B 




z 



E 



SB 



S 



a 



^ 



5 



^ 



ZuS 











Fig. 97. 



ROGERS' DRAWING AND DESIGN. 



55 



a 



Fig. 98. (See page 63.) 

Fig. 99 shows one of the devices in use for facilitating the laying out 
of letters ; this instrument is known as the lettering triangle and may be 
made of metal, hard rubber, celluloid, etc. 

The broken line a a contains lines of different inclinations by which 
the slanting parts of those letters, shown on the triangle, may be laid out. 

The highest inclined line may be used for the slanting strokes of the 
letter K ; the inclined line, situated next to the highest is intended to be 
used for the letters N, X and Y; the next inclined line is to be used for 
the drawing of the letters A, M and V and the lowest inclined line is 
used for the letter W. 

Other triangles and templates have been made for laying out lettering 
of different character, of which the example given is one of several in com- 
^ mon use. 




Fig. 99. 



56 



ROGERS' DRAWING AND DESIGN. 




Fig. 100. 



The above figure, loo, shows a pen made spec- 
ially for round writing- upon drawings ; while nearly 
all lettering is executed by a common writing pen 
this device deserves a description. 

The " nibs " of the pen are cut off, leaving a 
short, straight line at the point ; the width of this 
point is equal to the greatest thickness of a line 
which may be desired for the letters ; these pens 
are manufactured in various widths and numbers ; 
No. I is made for a stroke of about ys" in thick- 
ness ; the highest numbers are made with a point 
nearly like the ordinary writing pen point. 

The No. I pen may be used for capital letters, 
about one inch high and for small letters about }4" 
high ; the pen is always held parallel to the line ad 
at 45°, Fig. loo. 



A motion of the pen in the direction of the line 
a b produces a fine line ; all strokes, light or heavy, 
are made by means of the whole width and not by 
only one edge of the pen ; heavy strokes are made 
with the pen moving in the direction c d, Fig. lOo, 
with the whole width of the pen. The pen should 
move smoothly over the paper without any special 
pressure being brought to bear on it. 

Vertical strokes produced by a downward motion 
of the pen will not be quite so wide as the line c d. 

All strokes should be executed with an unaltered 
position of the pen "nibs," which must remain par- 
allel to the direction of the line ad and inclined 
about 45°. 

Letters containing circular curves are made with 
the pen in the same position ; a circle should not 
be made by one continuous motion of the pen ; it 
should be formed of two semicircles, taking care to 
smoothly join the two semicircles. 

' It is well for the beginner to lay out a number of 
squares in pencil and to practice the circular strokes 
within the squares ; the completed circle should be 
contained within the square. The light strokes 
will be parallel to the diagonal of the square, the 
vertical stroke should be parallel to the vertical 
sides of the square. The attractive appearance of 
the lettering will entirely depend upon the correct- 



ROGERS' DRAWING AND DESIGN. 



57 



ness of connecting the semicircles and straight 
lines of which the letters are composed. 

The size and thickness of the writing depend on 
the width of the pens and cannot be arbitrarily ex- 
ecuted by means of the same pen, without distort- 
ing the regular form of the characters. 

The pen must at all times be kept clean, as other- 
wise no clean-cut line can be obtained. The ink 
should be kept only on the outside or upper side 
of the pen, and its bottom side should be kept 
perfectly dry. As soon as the draughtsman notices 
that the bottom of the pen becomes wet he should 
cease writing with it, as it will produce an uneven 
line. 

The letters should be made with plain, even, 
clear-cut lines, well proportioned in all parts and 
especially well spaced. A special device, called an 
" inkholder," is used in order to keep a sufficient 
quantity of ink on the upper side of the pen. 

Free hand lettering should only be taken up after 
the student is proficient in mechanical lettering ; 
pencil guide-lines for letters and words should be 
drawn ; larger letters may first be penciled in very 
lightly, and an ordinary writing pen may be used 
for inking them in. 



A 3 CDEF'GMlJ^LJyrjVOP QRSTU VWXYZ 
/S3436?'890 

abcdey^hiJklrrirLopqrst-uvzoacyJZ 



COJ^J^ECTJJVGROD. 

Fig. 101. (See page e,3.) 

Letters should be so placed as not to interfere 
with the lines of the drawing and should clearly 
point out the part intended to be described. When 
single letters are used, they should be inked in be- 
fore the shade or section lines are drawn ; it is a 
good plan to start with the middle letter of the in- 
scription and work in both directions. 

The use of both the writing and drawing pen 
enables the lettering to be done in a much shorter 
time ; when the ruling pen is employed it is in con- 
nection with the tee-square and the set square. 

The appearance of a drawing will often be helped 
by a border put on in connection with the lettering. 



58 



ROGERS' DRAWING AND DESIGN. 



The four principal styles of letters used in mechan- 
ical drawing are Block, Roman, Old English and 
Script, each of which will be found illustrated under 



tendency is in the direction of simply designed 
letters, legibility being considered of vital impor- 
tance. 



ABCDEFGHIJKLMN 
O P QRST UVWXYZa. 

s^B Cn£:rGHI.JKLM 

NOPQRS TUVWXYZ 
abcdefghi/klmnopgrs t 

uvwxyz 

J234567a90i 1234567890^ 



Fig. 102. (See page es.) 



I 



this section of the work ; it will be found upon in- 
vestigation that most letters in use to-day are 
founded upon one of these four styles ; the modern 



It will familiarize the student with the standard 
alphabets in Roman, Block, Old English and Script 
to copy the several specimens given. 



ROGERS' DRAWING AND DESIGN. 



59 



The space between the nearest parts of all letters 
should be exactly alike ; this rule also applies to the 
space between each word ; between the words, of 
course, should be wider than the letter spaces. 



A reasonable space (never less than one-third the 
height of the letters used), should be left between 
lines of words. 






Fig. 103. (See paae os.) 



60 



ROGERS' DRAWING AND DESIGN. 



Mathematical accuracy should be aimed at as 
a rule in all lettering executed for mechanical 
drawings. A knowledge of punctuation, spelling, 
capitalization and paragraphing is essential in 



for the bottom and then the letters should be 
sketched with the utniost care ; the outlines may be 
ruled with a ruling pen, if desired, and the curved 
lines drawn with a compass ruling pen. 




Fig. 104. (See page r,s.) 



i.n.ffl.iv:y.vi.vn.Yni.K.x.ix.isx.xL.L.xc.c.D.M. 



^ 



■r 



Fia. 105. 



-5v. J'^ 



so , ■^oc JHo. -fcee- 



this work ; if unfamiliar with these subjects the 
student should acquire a. thorough knowledge of 
them. 

Perfectly horizontal ruled lines should first be 
drawn, one for the top of a line of letters, another 



The heavy or shaded stems of letters should all 
be of the same width ; after the outlines have been 
carefully penned in, the unfilled spaces may be 
''brushed" in with either liquid India ink or very 
black water color. 



ROGERS' DRAWING AND DESIGN. 



61 



3 
4 

3 



JBCDEFGHIJFCULrOPQRSTUF 

ffXYZ &. 

abcdef'gh ijklmnojjq rfstuvwxyz. 



6 

7 
S 
9 




Fig. lOB. 




at 34 



Fig. 107. 



62 



ROGERS' DRAWING AND DESIGN. 









''Mlf^ 



Fig. 108 



ROGERS' DRAWING AND DESIGN. 



63 



In Figures 96 and 97 
letters and numerals. 



are examples of block 



The squares are laid off in fine pencil lines and 
the desired letters may be sketched on the drawing 
in pencil and then inked in with pen and triangle ; 
when the inking is completed all the pencil lines are 
erased. This is a rudimentary form of letter that 
can be made with the aid of cross section paper. 

In Figs. 98 and 103 are shown two styles of free- 
hand lettering. The vertical letters are more diffi- 
cult to draw than the slanting ones. When making 
these letters two fine pencil lines should always be 
drawn, one at the top and one at the bottom of the 
letters and sometimes it is very convenient to rule 
a third guide line midway between the two others. 

These examples exhibit a form of lettering known 
as round writing ; the easy way to master it and 
its artistic appearance, combined with the rapidity 
with which it can be written, are its principal 
merits. 

In the upper part of Fig. 102, page 58, is shown 
another example of the block letter ; this is very 



distinct and readily executed by the aid of the 
drawing pen. In the lower part of the figure (102) 
are shown Italic letters and numerals; the proper 
angle for their slant is 23°. 



In Fig. 104 are shown ornamental letters based 
upon the Roman ; in the Roman letters the square 
is taken as the basis of construction; W takes the 
whole square, its height and width being equal ; I 
is one-quarter as wide ; A five-sixths, etc. 

In Fig. 105 are shown the form and proportions 
of the Roman numerals and their value in the Arabic 
method of expressing numbers. 

In Fig. 106 is given another example of Italic 
letters and numerals. 



In Figs. 107 and 108 are given illustrations of 
script letters and Figs. 109 and 1 10 will suggest to 
the student still other forms. 



m 



The letters shown in Fig. loi are constructed 
a simple form convenient for remarks, etc., 



needed to be placed in the margin of the drawing. 



64 



ROGERS' DRAWING AND DESIGN. 




ffl 1 



C^ 




I 2 



OD 



\P 



i 



<» 



s t 



5 



Fro. 109. 



[F 



(S) [^ § 
q] D° © 



i 



1^ ^^ ^ 






1 



d Da D J) K [L 
T 1 W M H V 



gj 




Od 




a 

D 




n 




Ds 


- 


It 




DD 




W 




\w 




n 


^ 


IG- 110. 




1 




© 




S) 


(D 





DQQ 



ROGERS' DRAWING AND DESIGN. 65 


SHADE LINES. 

In instrumental drawings shade lines are used for 
the purpose of making the reading of a drawing 
easier than if all lines were of the same thickness. 

By means of these, the draughtsman knows with- 
out referring to any other view of the object whether 
the part looked at is above or below the plane 
of the surface ; for instance, the rectangles in Fig. 
1 1 1 represent square projecting pieces, whereas the 


the plane of the paper, and also with all vertical and 
horizontal lines of the drawing, and to come from 
the upper left-hand corner of the drawing ; the 
direction of the light being indicated by the slanting 
edge of the 45° triangle as shown in Fig. 1 13. 

All the rays of light are not supposed to be 
emanating from one and the same point, but from 
a large and distant source of light and are thrown 
in parallel lines. 

The shade lines are the edges of such surfaces as 




































































rectangle 
differenc 

In ore 

uniform, 
assumed 
such a w 


Fio. 111. 


ales ; the 
de lines. 

may be 

light are 

ction, in 

rees with 












s m r 
e bein 

ler th; 
and t( 
to COl 
ay as 


t^ig. II 
y made 

it the 
3 avoid 
ne in a 
to mak 


2 reprc 
appan 

shading 
confus 
single 

2 an an 


isent sc 
;nt by t 

y on di 
ion the 
invaria 
gle of I 


uare h( 
he sha 

■awlngs 
rays of 
Die dire 
15 deg 


Fio. 112. 

" relief " and aid the reader or the student in under- 
standing the true character of the object with 
greater facility than could be done on drawings with 
all lines of one and the same thickness. 



66 



ROGERS' DRAWING AND DESIGN. 



The following rules should be strictly adhered to 
by the student in shading drawings : 

(fl). The rays of light are assumed to make an 
angle of 45 degrees with the plane of the paper and 
to come from the upper left-hand corner, at an 
angle of 45° with all horizontal and vertical lines as 
previously mentioned. 

(b). Each view of the subject should be consid- 
ered as a top view for the purpose of shading ; its 
top part will thus be exposed to the light. 

(c). Lines representing edges which cast shadows 
are to be drawn in heavy lines. 

(d). All the edges formed by the intersection of 
a light and dark surface or two dark surfaces, are to 
be shaded. 



Figs. 114 and 115 show two views of a square 
block or rectangular prism. In the top view, 
abed, the rays of light fall upon the rear side of 
the object, b c, upon the top, abed, and upon the 
left side, a b ; the light does not reach the front and 
right sides, a d, and, d c ; hence they are dark sur- 
faces ; the edges a d and d c, separating the light 
surface, abed, from the dark surfaces are there- 
fore shade lines. 

The explanations given in regard to the top view 
can also be applied to the other view of the same 
object and the lines e h and h g will thuS be ob- 
tained as shade lines. 




Fig. 113. 







ROGERS' DRAWING AND DESIGN. 67 








Figs. ii6 and 117 represent a hollow square- 




b 






prism ; its top view shows the shade lines a d and 
d c, that is, the right and bottom sides of the view 




c 










upon which the rays of light do not fall ; the lines 










b 








a 




d 




f 




? 








u 


Fig. U4. 










a 




A 




f 






Fui. 116. 




9 










i 




: m 







^ 




m 












r 


8 




M 


P 






n 










Fiu. iir. 1 










e f and f g are also shade lines for the same reason ; 




e 




h 


the vertical section r i p shows the right side p 










Fig. 115. 


and bottom r p, also the vertical line k 1, as shade 
lines. 



68 



ROGERS' DRAWING AND DESIGN. 




Fig. 118. 



Fig. 119. 



Another example of shade lines is given in Figs. 
ii8 and 119; in this case the top view shows a 
hexagonal hole, offering a good opportunity to con- 
sider the shading of objects with inclined planes. 
It will be plainly seen that the lines c d and b c 
must be shade lines and the lines a f and f e, situ- 
ated directly in the way of the rays of light, are 
light lines. 

The two remaining sides b a and d e of the 
hexagonal hole cannot readily be put down as 
shade or light lines. In order to find out^ their 
nature draw a number of 45" parallel lines oVer the 
figure in question and those faces of the hexagonal 
hole reached by the rays of light, represented by the 
45° lines. 

It will be seen that the edge b a intersects the ar- 
rows of light ; the face a b will therefore be a dark 
surface, and consequently must be shaded. It will 
also be noticed that the rays of light fall directly 
upon the face, an edge of which is the line e d, thus 
making this latter a light line. 

In order to illustrate more clearly the shading of 
lines belonging to planes, which are inclined in va- 
rious degrees to the direction of the light, the top 
view of a cube placed in three different positions is 
shown in Figs. 120, 121 and 122. In Fig. 120 the 
edge a b makes an angle of more than 45° with a 



ROGERS' DRAWING AND DESIGN. 



69 



horizontal line ; parallel lines drawn at an angle of 
45° representing the rays of the light will strike the 
side of which a b is an edge ; this side must there- 
fore be a light surface and the line will be a light one. 
In Fig. 12 1 thelineab forms an angle of 45 degrees 
with a horizontal line showing that the side of the 
cube of which a b is an edge is placed in a position 



than 45 degrees with a horizontal line. It will be 
observed that the side of which a b is an edge can- 
not receive any direct light, as the rays are broken 
by the edge of which b is the highest point; hence 
the edge b a is a shade line. 

In Fig. 123 are shown the front and bottom views 
of a cylinder ; the shade lines on the front are easily 




Fio. 120. 



Fig. 121. 




Fig. 122. 



parallel to the direction of the light ; for this reason 
said side is considered a ligrht surface and the line a b 
is a light line. This is done in every case in which 
the line in question forms an angle of 45 degrees 
with a horizontal line. The same is true for the 
line c d, which is parallel to a b. 

In Fig. 122 the edge a b forms an angle of less 



determined, in a similar manner to foregoing cases 
where a drawing has been made of a rectangle. 
Many draughtsmen, however, will only shade the 
plane sides of the cylinder, claiming that shade 
lines are intended to represent edges only; ac- 
cording to this view the bottom line of the eleva- 
tion of the cylinder should only be shaded. 



70 



ROGERS' DRAWING AND DESIGN, 



At times a tendency has been noticed to give the 
shade lines a more general application. As shade 
lines are always found on all right and bottom sides 
of those parts situated in front of the surrounding 
surface, it is very easy to recognize this condition 
in every similar case, whether the part in question 
be bounded by plane or by cylindrical surfaces. 

It is therefore recommended that shade lines be 
drawn in each case, independently of the character 
of surface, with very few exceptions, when a rigid 
adherence to this rule will tend to produce a bad 
effect. 

The shaded portion in the bottom view of the 
cylinder, Fig. 123, shows the manner in which circles 
are to be shaded when they represent projections of 
cylinders or circular holes. 

Fig. 124 shows that the circle is shaded between 
the points of tangency of the two 45-degree lines 
a b and f g ; the heaviest part of the shaded circle 
is near the 45-degree line c d e, passing through the 
center of the circle. The thickness of the lines 
should gradually decrease from e toward b and to- 
ward g ; to obtain the best result with neatness is 
to shift the center point of the compasses along the 
line c e, a distance equal to the thickness of the de- 
sired line. With the same radius used to describe 
the original circle describe now part of another 



circle, being careful not to run over the first one 
and to stop when the two lines coincide. 



I 




? 



Fig. 123. 



The shade line made in this way must not be 
drawn too heavy ; to assure the success of this op- 



ROGERS' DRAWING AND DESIGN. 



71 



eration it is necessary to have a very sharp needle 
point in the compass in order not to cause too large 
a hole in the center of the original circle. The 
shifting of the center may be avoided when 
drawing the shade line in the following manner: 
When the original circle is drawn keep the center- 




FlG. 121. 



point in the center and without changing the radius 
put the pen point in motion in the direction of the 
part of the circumference to be shaded. The pres- 
sure upon the pen is gradually increased as it ap- 
proa'ches the heaviest part of the shade line and 
then gradually diminished. 

Figs. 125 and 126 show the manner of shading a 



circular hole. The directions for this operation are 
the same as for shading the projections of a cylinder 
base, except that the opposite half of the circle is 




FiQS. 125 AND 126. 

shaded, which is done by shifting the center in the 
opposite direction. 



72 



ROGERS' DRAWING AND DESIGN. 



In order to make the conventional way of putting 
in shade lines more easily understood, a few illus- 
trations are added for the benefit of the student. 

In Fig. 127, the position of the shade lines in the 
top view is quite plain ; in the front view, Fig. 128, 
the bottom line a b of the smaller block placed in 



throws a shadow upori the corresponding part of the 
front face of the larger block and the line a b is 
therefore a shade line. Similar cases are given in 
Figs. 133 to 138. 

In the front view. Fig. 136, a portion a c of the 
bottom line a b is shaded and in Fig. 138 the whole 



a h 





1 


a h 





Ui^J 




i J i 



Figs. 127 and 128. 



Figs. 139 and 130. 



Figs. 131 and 132. 



the middle of the top face of the larger block is a 
light line. 

In Figs. 129 and 130, the small block placed on 
top of the larger one so that the front faces of both 
are in one plane shows a light line a b. 

In Figs. 131 and 132, the smaller block is moved 
forward so that its lower face a b is the front edge. 



of the bottom line a b is a shade line. 

Fig. 139 shows the front and top views of a prism. 
The shading of the top view does not present any 
new points. In the front view the face e b c f is 
dark ; the edge e b separating the light face c^a b c 
from the dark one and consequently e b must be a 
shade line ; e f is also a shade line for reasons ex- 



ROGERS' DRAWING AND DESIGN. 



73 




FiciS. 133 AND VU. 



Figs. 1.3.5 and 136. 



plained previously. The line b c of the upper base 
a c should also be shaded ; this would make part of 

the straight line a c 
heavy and the greater 
part light, which would 
produce a very odd 
effect and therefore the 
whole line, in similar 
cases, is drawn light. 
The line c f is made 
h e a V }' b )' many 
draughtsmen, as the 
surrounding of dark 
surfaces by heavy lines 

Figs. 1.37 and 138. 











L_ 


















a 


b 



brings out such surfaces much stronger. From the 
various examples given above it will be seen that 
no special rules can be given for shading, that is, 
rules that would cover all cases likely to arise.' 

The conventional practice introduces a great 
variety of exceptions to any rules designed for this 



7^ 

b i 

e 



Fid. 139. 



purpose. The draughtsman has to keep in mind 
the true purpose of putting in shade lines and place 
such lines where and whenever, in his opinion, they 
will serve as an aid to the understanding of the 
drawing. 



74 



ROGERS' DRAWING AND DESIGN, 



PARALLEL LINE SHADING. 

Plane surfaces are shaded by a number of par- 
allel lines running parallel to the length of the plane 
which is to be shaded. If the plane is to be repre- 
sented very light, it may be left blank or coveretl 
with very fine parallel lines, as shown in Fig. 140. A 
dark plane is shaded iiy a number of heavy parallel 
lines, Fig. 141. 



which does not receive any direct light. The heavy 
lines become lighter gradually and are drawn very 
fine near the midd'e of the cylinder; after this the 
lines are again dra.vn slightly heavier up to the side 
of the cylinder, which is nearest to the source of 
the lis>ht. The shadintr lines near the liofhter side 
of the cylinder should never be as heavy as the 
heaviest lines on the dark side of the cylinder ; this 
is illustrated in Figs. 143, 144 and 145. The surface 





Fig. 140. 



Fig. 141. 



Fio. 142. 



If the plane is parallel to the plane of the paper, 
the shading lines should be drawn with equal spaces 
between them throughout the full width of the plane. 
If the shaded plane is inclined to the plane of the 
paper it is shaded by a number of lines, with the 
spaces between these lines graciually increasing, 
while the thickness of the lines gradually decreases 
as may be seen in Fig. 142. 

A cylinder is shaded by a number of parallel lines, 
whjch are heaviest near to the side of the cylinder 



near the middle of the cylinder is often left blank, 
as it is difhcult to produce the effect of a light tint 
which is desirable at that place. 

A hollow cylinder or a concave surface is shaded 
similar to a cylinder, as shown in Fig. 146. 

The view of the sleeve nut shown in Fig. 147 
illustrates the manner in which conical surfaces are 
shaded. Some draughtsmen do this by drawing the 
shading lines parallel to the outside elements of the 
cone. A somewhat better result is produced, how- 



ROGERS' DRAWING AND DESIGN. 



75 




ever, by drawing the lines slanting and tapering to 
the vertex of the cone, virhich is to l^e shaded. 
Wherever possible an ordinary pin may be put into 
the board exactly in the vertex of the cone. Tlie 





Fir.. 146. 



ruling edge of the triangle is thus easily kept 
against the pin, securing the proper direction for 
the tapering shading lines. 



76 



ROGERS' DRAWING AND DESIGN. 



In Fig. 148 at a b is shown a cylinder placed in 
a horizontal position, which is slightly rounded at 
the end, so as not to have any sharp edge. This Is 
also indicated by shading lines drawn at right angles 
to the shading lines of the cylinder. 




FiQ. U" 




Fig. 148. 




Fig. 150. 




Fig. 151. 



ROGERS' DRAWING AND DESIGN. 



77 



Fig. 149 shows how this may be done by the aid 
of curved lines ; however the time .required for this 
method does not recommend it for ordinary work- 
ing drawings ; the same figure includes a spherical 
surface and shows how such surface may be shaded. 

Fig. 150 shows the shading of a curved cylinder. 

Fig. 151 shows a method of representation of 
knurled surfaces. The spacing of the inclined lines 
varies, being closer near the sides of the figure. 

In conclusion let it be noted that the best effects 
are, as a rule, produced by the fewest lines ; draw- 
ings executed to small scale will look best with a 
shading that does not include any very heavy lines ; 
larger scale drawings require the use of very heavy 
shading lines. 

In ordinary working drawings shading is, as a 
rule, but very little employed ; it is, however, some- 
times done to shade the surface of shafts and even 
bolts as well as other cylindrical parts of small 
diameter by a few conveniently placed lines. 



SECTION LINING. 

It is sometimes necessary to make use of a sec- 
tion in order that certain details, which would 
otherwise be hidden, may be shown in a plain. 




Fju. ir,:;. 




78 



ROGERS' DRAWING AND DESIGN. 



concise manner. The method used in shops, and 
the best for most purposes, consists of drawing 
parallel lines within the section, which lines are 
usually inclined 45 degrees. By changing the 
direction of these lines a clear distinction may be 
made between different pieces in the same view, 
which may be in contact. 

A difference in material is shown by a variation 
of the character of the sectioning, see Figs. 152 
and 153. The section lines are best drawn from 
left to right or from right to left, usually inclined 
45 degrees and about one-sixteenth inch apart. For 
large drawings the spaces between them may be as 
much as one-eighth inch. 

Placing the lines too near together makes the 
work of sectioning much harder ; the lines should 
not be drawn first in pencil, but only in ink, as the 
neat appearance of the drawing depends largely 
upon the uniformity of the lines in the section and 
these lines are to be spaced by the eye only. The 
process consists simply in ruling one line after 
another, sliding the triangle along the edge of the 
tee-square for an equal distance after drawing each 
section line. 

Figs. 154-162 inclusive, are examples of section 
lining quite generally used. 




Fig. 154. 




Fig. 155. 



ROGERS' DRAWING AND DESIGN. 



79 




Fig. lot). 




c 



Fig. 15s. 



Fig. 157. 




Fig. 159. 



80 



ROGERS' DRAWING AND DESIGN. 



Cast iron is indicated by a series of parallel lines 
of medium thickness, equally distant apart as shown 
in Fig. 154. 

Wrought iron is sectioned in the same manner 
as cast iron except that every alternate line is a 
heavy line, Fig. 155. 

Cast steel is sectioned by drawing two lines, of 
medium thickness close together, and the third line 
about one and one-half times as far from the 
second as the second is from the first and so 
on as shown in Fig. 156. 

Brass is sectioned by parallel lines similar 
to cast iron, except that every other line is 
broken; see Fig. 157. 

Babbit is sectioned like cast iron in both 
directions, forming little squares, Fig. 158. 

Wrought steel is sectioned by two light 
lines and one single heavy line. The light 
lines should be drawn similarly to those in Fig. 156 
for cast steel, and the heavy line should be about 
one and one-half times as far from the light lines as 
the distance between them, as shown in Fig. 159. 

Wooden beams are sectioned by a series of 
rines and radiating lines in imitation of the natural 
appearance of a cross section of an oak tree. Fig. 160. 



A beam or board is represented by lines run- 
ning similarly to those of the grain in an oak board. 
Fig. 160. 

Brick and stone are represented as shown in 
Figs. 161 and 162. 

Xbin strips of metal Vikc the stct'ion of boiler 
plates may be sectioned in the ordinary way by the 




Fig. 160. 



usual section lines ; but as this requires consider- 
able work and produces an ill effect in the drawing, 
It is often better to fill in the whole sectional area 
with solid black. 

In this case a white line must be left between the 
adjoining pieces ; this method is recommended only 
for small sections, see Figs. 163 and 164. 



ROGERS' DRAWING AND DESIGN. 



81 



w///////////////////mw//. 


w//mm//y/M//////////A 


%^.^%^^^^^%%^^ 


^m^^m^^M^MM^ 



Fig. 161. 




Fia. 162. 




Fig. 163. 





Fio. 164. 



s&. 



! 



lllllllMllltliiiii 



IIMIIIIM^^ ■— ^^^^^ 




e 



iiiiiiiiiiiiiiiiiiii 



GEOMETRICAL DRAWING. 



Geometry is the science of measurement ; it is the root from which all mechanical drawings 
issue ; the principles involved in the following problems, make up the fundamental bases of all instru- 
mental drawing, as well as all "laying out" of work in the shop, where great accuracy is required. 

The elementary conceptions of geometry relate to the simple properties of straight lines, circles, 
plain surfaces, solids bounded by plain surfaces, the sphere, the cylinder and the right cone. 

Higher geometry is that part of the science which treats of the relations of these to lines, circles, 
surfaces, etc. Some geometrical terms have already been described, to these are now added a few 
relating to the more advanced parts of this oldest and simplest of sciences. 

An axiom is a self-evident truth, not only too simple to require, but too simple to admit of 
dem.onstration 

A proposition is something which is either proposed to be done, or to be demonstrated, and is 
either a problem or a theorem. 

A problem is something proposed to be done. 

A theorem is something proposed to be demonstrated. 

A hypothesis is a supposition made with a view to draw from it some consequence which 
establishes the truth or falsehood of a proposition, or solves a problem. 

A lemma is something which is premised, or demonstrated, in order to render what follows 
more easy. 

A corollary is a consequent truth derived immediately from some preceding truth or demon- 
stration. 

A scholium is a remark or observation made upon something going before it. 
A postulate is a problem, the solution of which is self-evident. 



8.1 



86 ROGERS' DRAWING AND DESIGN. 

EXAMPLES OF POSTULATES. 

Let it be granted — 

I. That a straight line can be drawn from any one point to any other point ; 
n. That a straight line can be produced to any distance, or terminated at any point ; 
in. That the circumference of a circle can be described about any center, at any 
distance from that center. 

AXIOMS. 

L Things which are equal to the same thing are equal to each other. /, 

n. When equals are added to equals the two or more wholes are equal. [ 

in. When equals are taken from equals the remainders are equal. 

IV. When equals are added to unequals the wholes are unequal. 

V. When equals are taken from unequals the remainders are unequal. 

VI. Things which are double of the same thing, or equal things are equal to each other. 

VII. Things which are halves of the same thing, or of equal things, are equal to each other. 

VIII. The whole is greater than any of its parts. 

IX. Every whole is equal to all its parts taken together. 

X. Things which coincide, or fill the same space, are identical, or mutually equal in all 

their parts. 

XI. All right angles are equal to one another. 

XII. A straight line is the shortest distance between two points. 

XIII. Two straight lines cannot enclose a space. 

The tools used in geometrical drawing are the compass, with pencil and pen points, the ruling 
pen, straight edge and scales ; in the following pages will be found a series of exercises which have 
been selected with a view to their importance in their application in problems of accurate drawing. 



ROGERS' DRAWING AND DESIGN. 



87 



EXERQSES IN GEOMETRICAL DRAWING. 

To bisect a given straight line; that is, to divide 
it into two equal parts. 

Let AB be the given line, Fig. 165. 



/ 
/ 

/ 



/ 



B 



;c 



Fig. ]&5. 

From A as a center with a radius g-reater than 
one-half of the given line AB, describe the arc i 
2, From B as a center, and with the same radius, 
describe an arc, cutting the former at i and 2 ; 
then through the points of intersection draw the 



line 1C2 and it will divide the line AB into two 
equal parts at the point C. 

To bisect a given angle ; that is, to divide a given 
angle into two equal angles. 




Let ACB be the given angle. Fig. 166. 

With the vertex C as a center, and any radius, 
describe an arc cutting both sides of the given angle 
at I and 2. From i and 2 as centers, with any 
radius, describe arcs cutting each other at 3. 



88 



ROGERS' DRAWING AND DESIGN. 



Through this point of intersection draw the line 3C 
and it will bisect the angle as required. 

To divide a given angle into four equal parts. 
Let ACB be the given angle, Fig. 167. Bisect 




the given angle as described in Problem 2 by the 
line 3C. Bisect the angles 3CB and 3CA by the 
lines C4 and C5 and these lines divide the angle 
into four equal angles as required. 



To trisect a right angle ; that is, to divide it into 
three equal parts. 

Let ABC be a right angle, Fig. 168, that is, an 
angle with the sides perpendicular to each other. 
From B as a center with any radius, describe an arc 
cutting the sides of the angle at i and 4. 




B Fig. 16S. '4 *^ 

With the same radius and with 4 as a center, 
describe an arc cutting the former at 2. From i 
as a center with the same radius, cut the arc at 3. 

Through the points 2 and 3 draw the lines 2B 
and 3B and they will divide the angle into three 
equal parts as required. 



ROGERS' DRAWING AND DESIGN. 



89 



To draw a line perpendicular to a given straight 
line from a given point in that line ; that is, to erect a 
perpendicular to the given line at a given point in that 
line. 

Let AB be the given line and C the given point 
in that line, Fig. 169. 



3. 



B 



C 

FUi. IfiH. 



With any radius set off on each side of the point 
C, equal distances, as Ci and C2. From the points 
I and 2 as centers, with any radius greater than Ci 
or C2, describe arcs cutting each other at 3. 
Through the point of intersection draw the line 3C, 
which will be perpendicular to the line AB. 



To draw a perpendicular line to a straight line, 
from a given point without that line ; that is, to drop 
a perpendicular to a given line from a point with- 
out it. 

Let AB be the given line and C the given point, 
Fig. 170. 



1 ^^. 



\ 



\ 



/ 



D 



B 



f 



/ 

/ 



Fig. 170. 



From C as a center with any radius extending 
below the line AB describe an arc i 2, cutting AB 
at I and 2. From i and 2 as centers, with the same 
or any other equal radii, describe arcs cutting each 
other at 3. Through the point C and the point of 
intersection 3 draw the line 3DC ; then the line CD 
will be perpendicular to AB. 



90 



ROGERS' DRAWING AND DESIGN. 



To drop a perpendicular to a given line front a point 
which is nearly over the end of the line, Fig. i-ji. 

Let AB be the given line and C the given point. 
From any point i on the line AB as a center, with 
the radius iC describe the arc CE. 



\ 



/ 



\ 
\ 



\ 



V\ 



> 



/ 
/ 



/ 



Fig. 171. 



From any other point 2 on the line AB as a 
center, describe arcs cutting the former arc at C 
and E. Draw a line through the points C and E 
and the line CE will be the perpendicular required. 



Through a given point to draw a straight line 
parallel to a given straight line. 

Let AB be the given line and C the given point, 
Fig. 17:2. 

From C as a center with any radius describe the 
arc I, 2, cutting the line AB at 2. 



z' 



y' 



II 

\ 
\ 



\ 



7^ 



\ 



X 



\ 



\:i 



\ 



Fig. 172. 



With the same radius and 2 as a center, describe 
the arc C3. On the arc 2, i, set off from 2 the 
chord of the arc 3C, cutting it at i. Through the 
points C and i draw a straight line DiCE and it 
will be parallel to AB. 



ROGERS' DRAWING AND DESIGN. 



91 



To divide a straight line into any required number 
of equal parts (^say y equal parts\ 

Let AB be the given line, Fig. 173. 

From A draw a straight line AC forming any 
angle with AB and being of any length. Set the 
dividers to any convenient distance and set off 
seven equal divisions on the line AC beginning at 
A up to the point 7. 



6 



Jf 



>• 



/ / 



/ 

/ / 

•^ / / / ' 



.^ 






I 



I 
I 



I 
I 
I 



I 



X 



/ 



U 



Fig. 1T3. 



Join the points 7 and B by a straight line and 
draw parallels to it through the points i, 2, 3, 4, 
5, 6, and these lines will divide the given line AB 
into the required number of parts. 



To divide a given line AB into three and a half 
equal parts. 

Let AB be the given line, Fig. 174. 

Draw a line AC forming any angle with the 
given line AB. Upon AC set off 7 equal parts, be- 
ginning at A up to the point 7. 




— C 



Join the points 7 and B and through the alter- 
nate points, 5, 3, I, draw lines parallel to 7B. 
These lines will divide the given line AB into 3^ 
equal parts, as required. 



92 



ROGERS' DRAWING AND DESIGN. 



To draw upon a straight line an angle which shall 
be equal to a given angle. 

Let 1E2 be the given angle and AB the line 
upon which we intend to draw an angle equal to the 
given one, Fig. 175. 




B 



Fig. 175. 



From E as a center describe an arc i, 2, with any 
convenient radius. From any point on the line 
AB, say from C, as a center, and with the same 
radius describe the arc 3, 4. From 4 as a center, 
with a radius equal to i, 2, intersect the arc 4, 3, at 
3. A line drawn through the points 3 and C will 
form with the line AB the required angle. 



To construct an equilateral triangle, the length of 
a side being given. 

Let the straight line AB be the given side, Fig. 
176. 




B 



Fio. 176. 



From the points A and B with a radius equal to 
AB describe arcs cutting each other at C. Draw 
the lines AC and BC ; then will the triangle ABC 
be the required equilateral triangle. 



ROGERS' DRAWING AND DESIGN. 



93 



To cojisiruci an equilateral triangle, the vertical 
height or altittide being given. 

Let AB be the given vertical height, Fig. 177. 

Through the point B draw a straight Hne CD 
perpendicular to AB. 

Through the point A draw another straight line, 
EF, parallel to CD. From B as center with any 




convenient radius describe a semicircle cutting- CD 
at I and 4. From i and 4 as centers, with the 
same radius, cut the semicircle at 2 and 3. From 
B and through the points 2 and 3 draw the lines 
BG and BH ; then GBH will be the required 
triangle. 



To construct an isosceles triangle, with a base equal 
to a given straight line, and each of the two angles 
at the base equal to a given angle. 

Let D be the given line and E the given angle, 
Fig. 178. 




Fig. 178. 



Draw a line, AB, equal to the given line D. At 
the points A and B construct angles equal to the 
given angle E. Continue the sides of the angles 
until they meet at C ; then ABC will be the re- 
quired triangle. 



94 



ROGERS' DRAWING AND DESIGN. 



Two sides and the angle between them being given 
to construct the triangle. 

Let D and E be the two given lines equal re- 
spectively to two sides of the required triangle, 
and F the given angle, Fig. i 79. 



U 




Fig. 179. 



Draw a line, AB, equal to D, and at the point A 
construct an angle equal to F and make AC equal 
to E. Join the points C and B by a straight line, 
and ABC will then be the required triangle. 



Two sides and the (Lngle opposite one of them being 
given to construct a required triangle. 

Let D and E be the two given sides and let E be 
the side opposite which the angle is to be formed 
equal to F, Fig. 180. 




Fig. 181. 

Draw a line, AB, equal to D. At the point A 
form an angle equal to F. With the point B as a 
center and a radius equal to the given line E de- 
scribe an arc cutting AC at C. Join the points C 
and B. ABC is the required triangle. 



ROGERS' DRAWING AND DESIGN. 95 


To conslruct a square, the sides of which shall be 


To construct a square its diagonal being given. 


equal to a given line. (See definition, page 31.) 
Let AB be the given line, Fig. 181. 


Let BD be the given length of a diagonal, Fig. 
182. 


At the point A erect a perpendicular AD (see 


Bisect the diagonal BD at the point P by the 


page 89) equal in length to AB. 


straight line AC. 

1 


D 


\ 


c_ 






\ 
N 
^' \ 

\ 
\ 
\ 


/ 


k" 


. C 




/P\ 


\ 
\ 


y- 

\ 
\ 

\ 


A \ \ 


B 


\ 
\ 


/ \ 


/ 
/ 


\ 
\ 
X 


^AV '\ 


B 




1 

1 




Fu;. 1S2. 


Fig. 1»1. 




From the points B and D as centers, with a 




radius equal to AB, describe two arcs cutting each 


From P as a center with a radius equal to PB, or 


other at C. Connect D and C by a straight line 


PD, cut the line AC at the points A and C. Join 


and B and C by a straight line, and ABCD is 


the points AB, BC, CD and DA, and ABCD will 


the required square. 


be the required square. 



96 



ROGERS' DRAWING AND DESIGN. 



To construct a rectangle whose sides shall be equal 
to two given lines. (See definition, page 31.) 

Let AB and CD be the given lines, Fig. 183. 
Draw a straight line EF equal to AB, from E 
draw EH penpendicular to EF and equal to CD. 




D 



B 



Fig. 183. 



From H and F as centers with radii equal to AB 
and CD describe arcs intersecting at G. Join the 
points FG and HG; then EFGH is the required 
rectangle. 



To construct a parallelogram when the sides and 
one of the angles are given. (See definition, page 31.) 

Let AB and CD be the given sides and O the 
given angle, Fig. 184. 

Draw a straight line, EF, equal to AB. At E 
draw an angle equal to the given angle O. Make 
the side, HE, of this angle equal in length to CD. 




B 



Fig. 184. 



From the point F with a radius equal to CD 
and from H with AB as a radius describe arcs in- 
tersecting at G. Join HG and FG. EFGH is the 
required parallelogram. 



ROGERS' DRAWING AND DESIGN. 



97 



To construct a parallelogram when the sides and 
one of the diagonals are given. Fig. 185. 

Let CD be the given diagonal and AB and EF 



the lengths of the two sides. 




E- 
A- 



Fig. 185. 

Draw a line, GK, equal to the given diagonal 
CD. From G and K as centers, with radii equal in 
length to AB and EF describe arcs intersecting at 
L and H. Join GL, LK, KH and HG. GHKL 
is the required parallelogram. 



To find the center of a given arc, its radius being 
given. (See definition, page 33.) 

Let AB be the given arc and E the radius, Fig. 
186. 




Fig. 186. 



From any two points A and B on the given arc, 
as centers, with a distance equal to the radius E 
describe arcs intersecting at C ; then C will be the 
required center. 



98 



ROGERS' DRAWING AND DESIGN. 



To find the center and to describe the circle, three 
of whose points are given ; that 7S, to describe the 
circumference passing through three given points. 




Fig. 187. 



Let A, B and C be the given three points, Fig. 
187. 



With A, B and C as centers and any convenient 
radius, draw arcs cutting each other at D and E 
and at K and L, and through the points of their in- 
tersection draw lines KO and DO ; the intersection 
of these lines at O is the required center. With O 
as a center and OA as a radius, describe the re- 
quired circle. 




Fig. 188. 

To draw a tangent to a circle, passing through a 
given point on the circumference. (See definition, 
page 34.) 

Let A be the given point on the given circum- 
ference, Fig. 188. 

From A to the center O of the circle, draw the 
radius AO. Through A draw the line BC perpen- 
dicular to AO. The line BC is the required tangent. 



ROGERS' DRAWING AND DESIGN. 



99 



To draw a tangent to a circle from a given point 
without the circumference. 

Let A be the given point and C the center of the 
given circle. Fig. 189. 




Fin. 1X9. 



Join AC and bisect it at O. From O as center, 
with a radius equal to OC or OA describe a semi- 
circle, cutting the given circle at D. The required 
tangent is a line passing through A and D. 



To draw lines tangent to two given circles. 

Case I. — From O, the center of the larger circle, 
Fig. 190, draw any radius OE on which set off from 
E, a distance EG equal to the radius of the smaller 
circle. With O as a center and OG as radius de- 
.scribe the circle GHI and draw tangents PH and 




Fio. Ifll). 



PI to this circle from the center P of the other 
circle. (See the preceding problem.) 

From O and P draw perpendiculars to these tan- 
gents and continue them until they cut the given 
circles at AB and CD. Join the points. The lines 
AB and CD are the required tangents. 



100 



ROGERS' DRAWING AND DESIGN. 



To draw lines tangent to two given circles : 
Case II. — From O, the center of one of the given 
circles, Fig. 191, draw any radius OE and lengthen 
it outside of the circle up to G, making the distance 
EG equal to the radius of the other circle. 




From O as center and OG as radius, describe 
the circle GHI ; draw tangents PH and PI to this 
circle from the center P of the other circle. Draw 
perpendiculars to these tangents from O and P and 
they cut the given circles at the points A BCD. 
The lines joining the points A and B and C and 
D are the required tangents. 



To inscribe a square in a given circle; that is, to 
draiv a square within the circle, with all the vertices 
of its angles resting on the circumference. 

Let ABCD be the given circle. Fig. 192. 



B 




Draw two diameters, AC and BD, at right angles 
to each other. Draw the lines AB, BC, CD, and 
DA, joining the points of intersection of these di- 
ameters with the circumference of the circle ACBD. 
ACBD is the required square. 



ROGERS' DRAWING AND DESIGN. 



101 



To describe a square about a given circle. 

Let EGHF be the given circle, Fig. 193. 
Draw two diameters, FG and EH, at right angles 
to each other. At the points EGHF, where these 




F 

Fig. 193. 



diameters intersect the circumference of the given 
circle draw lines perpendicular to these diameters. 
These lines will intersect each other at ABCD. 
which is the required square. 



To inscribe a hexagon in a given circle. (See 
definition, page 32.) 

Draw a diameter AB in the given circle, Fig. 194. 

From A and B as centers, with a radius equal to 
the radius of the given circle, describe four arcs 
cutting the circumference of the circle at DEF and 
G. Join these points by straight lines. ADEBFG 
is the required hexagon. 




B 



G F 

Fig. 194. 

To divide the circumference of the circle into six 
equal parts. 

We set the dividers to equal the radius of the 
circle and get the required result by stepping the 
radius six times around the circle. 



102 



ROGERS' DRAWING AND DESIGN. 



To construct a hexagon upon a given line. 

Let AB be the given line and let it equal in 
length a side of the required hexagon, Fig. 195. 

From A and B as centers describe arcs cutting 
each other at G, the radii of the arcs being equal to 
AB. 




Fig. 195. 

From G as center with the same radius de- 
scribe a circle. With the same radius set off arcs 
cutting the circumference at CEF and D. Join 
these points by straight lines and they will form the 
sides of the required hexagon. 



To describe an octagon in a given square. (See 
definition, page 32.) 

Let ABCD be the given square, Fig. 196. 
Draw the diagonals of the square cutting at E. 




Fig. 19K. 



From ABC and D as centers, with a radius AE, 
describe arcs cutting the sides at GH, etc. Join 
the points so found to complete the required 
octagon. 



ROGERS' DRAWING AND DESIGN. 



103 



To describe an octagon on a given litte, one side of 
the octagon being given. 

Let AB be the given side, Fig. 197. 

Lengthen the line AB both ways. Erect perpen- 
diculars to this line at A and B. 




Fig. 197 



Bisect the external angle at A by the line AH, 
and the external angle at B by the line BC. Make 
AH and BC equal to AB. Draw HG and CD par- 
allel to AE and equal to AB. 

From G as center, with a radius equal to AB, cut 



the perpendicular AE at E, and from D as center 
with the same radius cut the perpendicular BE at F. 
Complete the octagon by joining GEE and D. 

To draw a regular polygon of any number of sides 
on a given line. (See definition, page 30.) 

Let C5 be the given side of the required poly- 
gon, Fig. 198. "n /p 




o c 

Fig. 19S. 

Lengthen the line C5 to O. With C as center 
and a radius equal to C5 describe the semicircle 
O I 2345, and divide this into as many equal parts as 
there are sides in the required polygon. Join C 
with 2, 3, 4, etc., by straight lines. With 2 as a 
center and a radius equal to C5 describe an arc 
cutting the line C3 at D. With D as center, and 
with the same radius draw an arc cutting the line 
C4 at E, and so on. Join the points C2D, etc., to 
form the required polygon. 



104 



ROGERS' DRAWING AND DESIGN. 



To inscribe a regular pentagon in a given circle. 
(See definition, page 32.) 

Draw two diameters AC and DB at right angles 
to each other, Fig. 199. 

Bisect the radius OB at I. With I as center and 
a radius equal to I A describe an arc cutting the di- 
ameter DB at J. 




Fio. 199. 



A straight line joining A and J is equal to one 
side of the required pentagon. With arcs of a 
radius equal to AJ set off on the circumference the 
points where the sides of the pentagon will ter- 
minate. 



To inscribe a regular polygon of any number of 
sides, within a given circle. 




Fig. 200. 



Draw two diameters AC and D7 within the given 
circle. Fig. 200, at right angles to each other. 



ROGERS' DRAWING AND DESIGN. 



105 



Divide the diameter D7 into as many equal parts 
as there are sides in the required polygon. Let it 
be seven in this case, at the points 123456. 

Lengthen the diameter AC making AK equal 
to three-fourths of the radius of the given circle. 
Through K and 2 draw a straight line cutting the 
circumference at L Join the points D and I by a 
straight line, and it is equal in length to one side 
of the required polygon. Set the dividers to equal 
this side, and set off the other sides around the cir- 
cumference. 



To describe an octagon in a circle. 

Draw two diameters at right angles ; these diam- 
eters divide the circumference into four equal arcs. 
Bisect these arcs to complete the octagon. 



To drazv an oval by circular arcs. 

Let CD be the major axis and AB the minor 
axis of the oval, Fig. 201. 

Find the difference of the semi-axes and set it off 
from O to e and f on CD and AB. Bisect ef and 
set off one-half of it from e to g and draw gh 
parallel to ef. 



From the center O on CD lay off the distance 
Oi equal to Og and draw hi ; through the points i 
and g draw the lines Ri and gR parallel to gh and 
hi. With Cg as a radius and the points g and i 
as centers, draw the arcs jCm and nDp ; with RA 




as a radius and R and h as centers draw the arcs 
jAn and mBp meeting the small arcs in the points 
j and n and m and p. The figure AnDpBmCj 
is the required oval. 



106 ROGERS' DRAWING AND DESIGN. 




MECHANICAL METHOD. 

Draw a line AB equal to the major axis of the 




y^ "^ 


^^\ 






/ ^ 


^ \ 




required ellipse, Fig. 202. 


% 


^^^--'''^ 




Bisect the line AB at E. At E draw a line CD 




\ 




perpendicular to AB. Make ED equal to EC and 




A / 


\ 




equal to one-half the minor axis. Set the compass 




/ \ ^^ 


\ 




to a distance equal to AE or EB, and with C or D 




/ \ ^^ 


\ 




as center, describe an arc cutting the major axis at 




/ A 


\ 




F and G. F and G are the foci of the ellipse. 




// \ 


\ 




Fasten the ends of a string, whose length is equal 




/ \ 


^ 




to the length of the major axis, AB, at thfe foci F 


to 


/ \ 


^ 


and G. This may be done by fixing pins at the 
foci and by providing the ends of the strings with 


^^ \ 






""x \ 






loops. 




\ ""^N \ 


, 




Trace a curve with the point of a pencil H 




\ ^^ \ 


/ 




pressed against the string so as to keep it stretched. 




\ ^^^ \ 


/ 




The curve thus traced will be the required ellipse. 




\ ^'1 


/ 




GEOMETRICAL METHOD. 




^ / 




Draw a rectangle ABCD enclosing the axes of 




\ ^ 


r» / 




the ellipse, Fig. 203. 




\. 


y^ 




Let EF be the major axis and HJ the minor 




^^"■-^^■^ 


^^^.^ 




axis. Divide AB, DC and EF into a like number 
of equal parts making the number an even one. 




Fir.. 202. 






The greater the number the more accurate will be 


To draw an ellipse, the major and the minor axes 


the resultant ellipse. Let the number in this case 


being given. (See definition, page 36.) 


be 8. 



ROGERS' DRAWING AND DESIGN. 



107 




From 2, 4, 6, and from corresponding points in 
DF draw lines to H. From tlie points placed on 
KB and FC draw lines to J. From J and H draw 
lines through 5, 3 and i and through correspond- 
ing points on LF to meet those already drawn. 
Through the intersection of 2H with Ji, 4H with 
J3, etc., draw the outline of the ellipse. Finish 
carefully in pencil, freehand, and then ink in with 
aid of an irregular curve. 

To describe a parabola, the base BC and the alti- 
tude EF being given. (See definition, page 36.) 



On the given line BC construct the rectangle 
ABCD with an altitude or height EF, Fig. 204. 

From F the middle point in BC erect the perpen- 
dicular EF ; divide AB and BF into the same num- 
ber of equal parts, say four. In like manner divide 
DC and FC. Draw lines from 2, 4 and 6 on AB 
and from corresponding points on DC to E ; from 
5, 3 and I and from corresponding points in FC 
draw lines parallel to EF, meeting the lines drawn 
to E from 2, 4, 6, etc. 

Through the intersection of 5 with 6, 3 with 4 
and I with 2 and corresponding ooints, draw the 
curve of the parabola. 




108 



ROGERS' DRAWING AND DESIGN. 



To describe a hyperbola, the transverse axis, the 
altitude and the base being given. (See definition, 
page 36.) 

Let FI be the axis of the hyperbola, EI its alti- 
tude and BC its base, Fig. 205. 



F 



///I 1\\ 
//// >\\ 

/'/111 ' \ \ 



1 1 , \ 



/ ' ' i I I \ \ \ 

/ ' ' ' ^ \ 




5 3 1 E 

Fig. 20.5. 

On BC construct a rectangle ABCD with EI as 
its altitude. 

Divide AB and BE into the same number cf 
equal parts, say 5. Divide DC and EC in like 



manner. From F draw lines to the points of divi- 
sion on BC. From the points of division on AB 
and DC draw lines to I. 

Through the intersection of 8 with 7, 6 with 5 
and corresponding points, draw the curve of the 
hyperbola. 




Fig. 20fi. 

To construct a spiral composed of arcs of various 
radii. 

Let ABC be a small equilateral triangle, Fig. 206. 

Note — A spiral is a curve described about a fixed point, and which 
makes any number of revolutions around that point without returning 
into itself. 



ROGERS' DRAWING AND DESIGN. 



109 



Lengthen the sides AB, BC and CA. With B 
as a center, BA as a radius, describe the arc AG 
meeting the line BC prolonged at G. With C as 
center and CG as radius, describe the arc GE meet- 
ing the line AC prolonged at E. With A as center 
and AE as radius, describe the arc EF meeting BA 
prolonged at F, and so on, using successively the 
points BCA for centers. 

By using any regular polygon in the same man- 
ner, that is, lengthening its sides and taking the 
angular points of such figure for centers success- 
ively in order, as in the above problem, a different 
spiral may be formed. 



To draw the outline of a snail by circular arcs. 

Let C be the axis or center of rotation upon 
which the snail is fixed, Fig. 207. The point B 
nearest to the center and the point A most distant 
from the center being also given. 

From the center C describe a circle whose diam- 
eter shall be equal to one-third of AB and divide 
the circumference into any number of equal parts, 



as I, 



3. 4. etc. 



Draw through each of these points tangents to 
this circle. Then from the point i as center, lA as 
radius, draw the arc 1-2' and from 2 as center, 



2-2' as radius, describe the arc 2-3'; and from 3 
draw the arc 3-4' and so on, taking in order the 
points I, 2, 3, 4, etc., as centers. 




Note. — The .<;nail is a mechanical movement u.secl for a great variety 
of purposes, as in time-pieces, rlrop ii\otions, etc. 



110 



ROGERS' DRAWING AND DESIGN. 



To draw the outline of a heart-wheel. 




6 

Fig. SOS. 
No'rE. — The heart-wlieel is a popular mechanical device producing 
uniform reciprocating motion. 



Let C be the axis or center of rotation, upon 
which the heart-wheel is fixed, Fig. 208, and let AB 
be the required extent of the rectilinear motion, 
A being the nearest point to the center and B the 
most distant. 

From the center C with a radius equal to 
CB describe a circle. Divide this circle into 
any number of equal parts, say 12, and 
through the points of division draw radii Ci, 
C2, C3, C4, etc. 

Divide the line AB into half the number of 
equal parts, the circle is divided into (in this 
case six), as i, 2, 3, etc. Then from the cen- 
ter C with the distance Ci on the line AB, 
describe an arc cutting the first radius at the 
point D ; then take the other divisions on the 
line AB and in succession with them from 
the center C draw arcs, cutting their respec- 
tive radii Ci, C2, C3, etc., at the points DEFG 
and H, which are the points in the required 
heart-wheel curve, its highest point being C 
and its lowest A. 
The construction of various machine parts in- 
volves many problems similar to the preceding ; 
these will be introduced when treating of the design 
of mechanical motion and the construction of parts 
of various machines. 



ISOMETRIC, CABINET AND ORTHOGRAPHIC PROJECTIONS 

AND 

DEVELOPMENT OF SURFACES. 



The word projection means to throw forward, and in mechanical drawing it is the projecting or 
throwing forward of one view from another ; in drawings the lines in one view or plan may by this 
system be used to find those of others of the same object, and also to find their shape or curvature as 
they would appear in other representations. 

Isometric projection is that in which but a single plane of projection is used. 

Cabinet projection is somewhat like isometric projection; the cabinet projections are : i, a 
horizontal line ; 2, a vertical line and 3, a 45-degree line ; all measurements on the drawing must be laid 
off parallel to these axes ; cabinet projection is one of several systems of oblique projection. 

Orthographic projection. The primary geometrical meaning of the word orthographic is 
''of or pertaining to right lines or angles," hence all the projecting lines are either horizontal or vertical. 

Drawings made up in this manner will be easily understood by many people unacquainted with 
the special methods of drawing generally used in mechanical branches. 

Development of surfaces will be defined and illustrated under its own chapter, page 162. 

Objects represented as thus described give a clear understanding of all their dimensions, and 
approximately show them as they appear to the eye of the observer ; the method of representing objects 
as they really appear to the e\ e is called perspective drawing. This latter method, however, presents 
so many difficulties of construction, that various other means have been devised, all aiming to give 
the advantages of perspective, and avoiding at the same time the difficulties of construction. These 
methods, also called false perspective, are described under the heading of isometric projection, and 
will be further explained in the following chapter under the title " Cabinet Projection." 



113 



114 



ROGERS' DRAWING AND DESIGN. 



ISOMETRIC PROJECTION. 

Figure 209 shows a solid figure, a cube, with 
equal sides and resting on one of its corners ; the 
lines ac, ab and ag are called isometric axes ; these 
axes form an angle of 1 20 degrees with each other. 




Fig. a». 



They may be drawn by the 30° and 60° triangles ; 
the lines ac and ab forming angles of 30° with a 
horizontal line ; ag is a vertical line. 

All the lines in this figure are parallel to these 
axes, viz.: all the lengths are parallel to ab and all 
the widths are parallel to ac. 




Fig. 210. 



ROGERS' DRAWING AND DESIGN. 



115 



The method of thus representing objects is called 
isometric projectioti ; drawings made in this manner 
show very clearly, with one view, the object as it 
appears when looked upon ; all the sizes of the 
object are drawn full size, or made to one scale, 
parall(;l to the isometric axes. 

With these rules in mind several objects will be 
represented in isometric projection in order to ex- 
plain its principles. 

To draw a square block, 4 by 2" by 2" , Fig. 210. 

First draw the isometric axes, ab, ac and ad ; ab 
is a vertical line whereas ac and ad are lines form- 
ing angles of 30° with a horizontal line ; make ab 
equal to 2 inches, ad equal to 4 inches and ac equa 
to 2 inches; from c draw cf, parallel and equal to 
ab, and from d, draw dh, also parallel and equal 
to ab. 

Join the points b and f ; and the line bf will 
be equal and parallel to ac. Then join the 
points b and h and the line bh will be equal and 
parallel to ad ; from the point f, draw the line fg 
equal and parallel to bh, then draw the line gh, 
which will be equal and parallel to bf ; from the 
point of intersection g draw the vertical line gk, 
from c and d draw the lines cb and dk, respect- 
ively, and parallel to ad and ac. 



To draw a rectangular frame made 0/ wood y^' 
thick, the outside dimensions being 16" long, 8" wide 
and 2" deep, as shown in Fig. 211. 

First draw the isometric axes ab, ad and ac : make 
the line ab equal to the depth required, or 2", the 
line ad equal to 16" or the length desired for the 
frame, and finally the line ac equal to 8' or the 
width. 




FlG.I-'ll. 



116 



ROGERS' DRAWING AND DESIGN. 



Now draw the lines cf and de equal and parallel 
to ab and then draw the lines fb and eb, equal and 
parallel to ac and ad, respectively. From the point 
f draw ft equal and parallel to ad. 

Next join the points e and t and the line et will 
be parallel and equal to ac. 

Now make mb and bn, tq and tr, each 
equal to ^" for the thickness required 
in this example ; draw the lines 
mp and gr parallel to eb and 
also draw the lines qk 
and hn parallel to bf. 
The two lines gr and 
qk intersect at s ; from 
this point s, draw a 
vertical linesu, parallel 
and equal to ab. 

From u draw a line 
parallel to ad in the 
direction of ac and 
cutting this latter line ; 
also draw from u a line 
parallel to ac in the 
direction of mp and 
cutting said line. 

Note. — For objects as represented in figures 210 and 211 an iso- 
metric projection is desirable, but when the objects to be drawn contain 
curved surface lines the application of the above described method is 
limited. 




Fig. 212. 



To draw a right cylinder in a horizontal position, 
as shown in Fig. 212. 

Draw a square abed, 
Fig. 213, with sides of 
exactly the same 
length as the diameter 
of a circle whose sur- 
face is to be the base 
of the required cylin- 
der. Within this 
square draw ' a circle 
efgh, tangent to the 
square and its diame- 
ter equal to that of the 
base of the cylinder. 
Next draw the diagonals ad 
and be, cutting the circle at the 
points eghf ; join the points e and g, 
g and h, h and f, e and f by straight lines 
and extend these lines until they meet the 
sides of the square. 

These lines cut off equal lengths of the sides of 
the square in its four corners, so that ai^a2=d3^ 
d4, etc. 

Now suppose that the required cylinder is placed 
in a square prism, so as to exactly enclose the cyl- 
inder as shown in Fig. 214. 



ROGERS' DRAWING AND DESIGN. 



117 



It is evident that the prism will have two ends equal to the square shown 
in Fig-. 213, and that the length of the prism will be equal to that of 
the cylinder. 

Then draw the prism in isometric projection as explained 
on page 115; draw the diagonals AD and BC in the end 
ABCD of the prism and lay out the isometric pro- 
jection of the circle which is to form the base 
of the required cylinder. Set off on the 
line AC a distance Ai equal to ai in 

o I 3 A C, 



m 



\ 






/ 


/ 




/h 


N 


V 




\f 


) 


/ 






\ 



n 



a 



k 

Fio.213. 



Fig. 213; from the point D on the line 
DB, set off the distance D4, equal to 
ai, in Fig. 213, or equal to Ai in Fig. 
214. 




118 



ROGERS' DRAWING AND DESIGN. 



Through the point i draw a line parallel to AB, 
cutting the diagonals AD and BC at the points e 
and f ; through the point 4 draw also a line parallel 
to CD intersecting the diagonals at the points g 
and h ; draw the line mn through o, parallel to AB, 
cutting AC at m and BD at n ; draw a line parallel 
to AC through the same point O, cutting the line 
CD at 1 and the line AB at k. 

The points k, e, m, g, 1, h, n and f are points 
through which the required circle drawn in isometric 
projection will pass. The curve thus obtained is 
evidently not a circle, but has the form of an ellipse, 
its minor axis being eh and its major axis fg. This 
ellipse may be drawn by any method explained in 
the section pertaining to Geometrical Drawing. 

The other end of the cylinder, which is to be in- 
scribed in the figure KLMN, may be drawn in the 
same manner as already explained, and the ellipse 
GF will be obtained. 

In order to complete the isometric projection of 
the cylinder draw the lines Gg and Ff, joining both 
faces of the cylinder ; these lines are to be drawn 
through the ends of the major axis of both ellipses 
and they are tangent to these two curves. 

To draw a pattern of a crank, shown in Fig. 21^, 
isometric projection. 

The pattern consists of two cylinders joined by a 



board. The larger cylinder into which the shaft 
will fit is 3" in diameter and 25^" long; the smaller 
cylinder to which the crank pin is to be fitted, is 2" 
in diameter and 2 " long. The distance between the 
center lines of the two cylinders is 5". 

Proceed as follows : 

Describe a circle 3" in diameter, as in Fig. 216, 
and draw a square around it, and within the square 
draw two diagonals and other lines as in Fig. 213 ; 
draw the isometric projection of a prism having 
Fig. 216 as a base and a length equal to 21^"; said 
prism is marked ABCDdab and its hiddqn parts 
are not shown. 

In this prism lay out the isometric projection of 
the larger cylinder, whose front face will be the 
ellipse klNcjM. 

Fig. 218 shows only a small part of the ellipse 
forming the rear end of the cylinder and this small 
visible part is represented by mi. 

Through the center of the first ellipse draw the 
line MN parallel to CD and the line kc parallel to 
AD ; then draw the line eg through the point c and 
parallel to Aa and equal to t^/^". 

The point g indicates the place where the board, 
connecting both cylinders, is fastened to the first 
cylinder. The board intersects the cylinder, form- 
ing an additional ellipse, or more properly, a part 



ROGERS' DRAWING AND DESIGN. 



119 



of an ellipse, represented in Fig. 21S by uge ; this part of the 
ellipse is exactly equal to the part jcN of the ellipse McNk, 

— -f — 



■"»;" 




/•- 



Fm. 217 



.i.._ 




M 



r^ zi'- 



FlQ. 216. 



Fio. 215. 



and may be constructed by drawing from different points of 
the curve, jcN, a number of lines parallel and equal to eg. 




Fio. 218. 



120 



ROGERS' DRAWING AND DESIGN. 




The line uj is a tangent to both of these curves. 

From the point g draw the Hne gf parallel to ck and equal to 
2^'; through f draw the line hp parallel to CD so that hf is 
equal to fp, each of these being equal to one inch ; from the line 
hp draw the isometric projection of the prism hpsnto, which is 
to enclose the smaller cylinder. 

The base of the latter is shown in Fig, 
in Fig. 218 by the ellipse vfvv. The 
length of the prism is to be equal to 2". 

When the small cylinder has been 
drawn in isometric projection within this 
prism draw the line vw through the cen- 
ter of the ellipse vfw and parallel to hp ; 
draw the line vr through the point v, the 
distance vr being made equal to one 
inch and through the point r draw the 
line rm, tangent to the ellipse mi. 

The lines wu and ve are both tangent 
to the ellipse uge. The hidden parts of 
the object are not indicated in Fig. 218. 

Fig. 219 represents a tool chest drawn 
in isometric projection It is given here 
as an example of a large class of objects 
well adapted for representation by this 
method. 



ROGERS' DRAWING AND DESIGN. 



121 



CABINET PROJECTION. 

Cabinet Projection is somewhat similar to Iso- 
metric Projection ; its difference consists in selecting 
three axes to which all measurements of the object 
are drawn parallel ; see a, b, c, following : 




Fin. -iSS. 



The axes for cabinet projection are : i, 



ime; anc 



a 45' 



a hori- 
line, as 



zontal line ; 2, a vertica 
shown in Fig. above. 

It is to be remembered that : 

a. All horizontal measurements, parallel to the 



length of the object must be laid off parallel to the 
horizontal axis, in their actual sizes. 

b. All vertical measurements, parallel to the 
height of the object, must be draAvn parallel to the 
vertical axis, in their actual sizes. 




Fig. 221. 

c. All measurements parallel to the thickness of 
the object must be laid off on lines parallel to the 
45° axis, in sizes of only one-half of the actual cor- 
responding measurements. 

It is not essential which side of the object should 
be considered its length and which side its thickness. 



122 



ROGERS' DRAWING AND DESIGN. 



To draw a cube in cabinet projection, as^hown in 

Fig. 221. 

Suppose each side of the cube to be 3" long. 

Draw the three axes: ab=horizontal axis, bd^ 
vertical axis, and bc=axis inclined 45°. On the line 
ab set off, from the point b, the distance b]=3"; on 
the line bd, from b, lay off b2^3"; and on the line 
be measure off f'^ij^". A vertical line drawn 




Fig. 222. 



through point i parallel to the vertical axis bd and 
a horizontal line drawn through point 2 parallel to 
the horizontal axis ab will intersect at point 3 and 
thus complete one face of the cube b-2-3-1. 

Now, through the point 4 draw a vertical line 
parallel to bd and through point 2 draw a line in- 
clined 45° with the horizontal ; these two lines 



intersect at the point 5 and complete the side b-2-5-4 
of the cube. 

The remaining lines are drawn in a similar man- 
ner, parallel to the axis, from the points 3 and 5, 
intersecting at the point 6 and showing the top of 
the cube 3-6-2-5. 




Fig. 223. 



Next draw through tlie point 4 a horizontal line 
and parallel to ab and through the point i a line in- 
clined at 45° and parallel to be ; these two lines cut 
at the point 7 ; join the points 6 and 7 by a straight 
line and cube is complete. 



ROGERS' DRAWING AND DESIGN. 



123 



In Fig. 222 is shown in cabinet projection the 
frame represented in Fig. 211. The length of the 
frame, 16 inches in actual measurement, is repre- 
sented here on the 45° axis by only one-half of its 
actual size or 8 inches long ; all the 
other measurements are equal to the 
actual sizes of the object, as described 
on page 122. 



zontal axis ; 3, parallel to the vertical axis, that is, 
in a standing position. 

The first position of the cylinder being the most 
convenient for drawing it in cabinet projection ; it 
will be considered here before the others. 



m 





(e 


v 


\ 


\ 

\ 


^ 


p 


( 



n 

Fig. 224. 



B 




Fio. a'). 



To draw a right cylinder in cabinet projectioti, its 
base to be the circle shown in Fig. 22^. 

The cylinder may be placed in the following po- 
sitions : 

I, parallel to the 45° axis; 2, parallel to the hori- 



ExAMPLE I. — Draw in cabijiet projection, the 
prism abcgfedh. Fig. 22 j, enclosing the cylinder ; the 
face of the prism, abcg, will contain the visible base 
of the cylinder; which is shown in Fig. 223 by the 
circle kl, which is equal to it. 



\^ 



124 



ROGERS' DRAWING AND DESIGN. 



In the rear end of the prism draw the circle nm 
for the other end of the cylinder, and draw the lines 
kn and Im tangent to both circles ; this completes 
the cabinet projection of the cylinder. 

It is advisable to select this position for all cylin- 
ders, as much as possible, when they are to be 
drawn in cabinet projection, as the circles repre- 
senting the faces of the cylinder may be drawn by 
circles and the dra\/ing of ellipses is avoided. 

Example II. — Describe the circle forming the base 
of the required cylinder. Fig. 22^, within the square 
ABDG, Fig. 22^, and draw ihe diagonals BG and 
AD, cutting the circle at the points hefg. 

Through the points e and h draw the line ehi and 
through the points fg the \\n^ fg2 ; the distance Ai 
will be equal to the distance B2. 

Now, . assuming that the cylinder is contained 
within a rectangular prism, each end of which is 
equal to the square shown in Fig. 224 and the 
length of which is equal to that of the cylinder, 
draw this prism in cabinet projection as shown in 
Fig. 225. 

Lay out the axes ab, bd and be ; make ab equal 
to the length of the prism, that is, equal to the 
length of the required cylinder; make bd equal to 
AG, Fig. 224, and be equal to }4 of the distance 
AB In Fig. 224. 



Through the point c draw a vertical line ce equal 
and parallel to bd, then join the points d and e by 
a straicfht line thus forming- the figure bdec, which 
will be one end of the prism ; from the points a and 
f draw the lines ah and fg, each equal and parallel to 
be ; then -draw the line gh equal and parallel to af. 

The figure afgh thus obtained is the other end of 
the prism. 

Now, lay out one face of the cylinder within 
bdec. In order to do this draw the diagonals dc 
and eb, set off from b on the line be the distance bi 
equal to one-half of the distance Ai in Fig. 224 and 
on the same line be, Fig. 225, point off the distance 
2c from the point c and equal to bi. 

Through the points i and 2 draw vertical lines 
which will intersect the diagonals eb and cd ; the 
points of intersection thus obtained together with 
the points 4, 5, 3 and 6 — it is evident how these 
points are found — define the position of the 
curve which will represent the circle forming 
one face of the cylinder as it appears in cabinet 
projection. 

The curve within afgh is to be drawn in a similar 
manner for the other end of the cylinder. Two 
horizontal lines, each tangent to both these 
curves, will complete the cabinet projection of 
the cylinder. 



ROGERS' DRAWING AND DESIGN, 



125 



Example III. — From the drawing in this figure 
226, it is evident that the construction in this case is 
exactly the same as in case 2. 

From the above problems it will be seen that 





v 




\ 


:---^,<" 


^ 


^=::^^^ 


==^ 


\ 






•^^ 


\ 


y 


^ 





Fig. 226. 



objects with circular forms which are to be drawn 
in cabinet projection should be placed preferably 
with all or most of its circles as in the cylinder rep- 




PiG. 23T. 



resented in Example I ; in this position, as already 
previously stated, all circles in the object will be 
represented by their actual sizes in the cabinet pro- 



126 



ROGERS' DRAWING AND DESIGN 




jection and in this manner the construction of diffi- 
cult curves may be avoided. 

Isometric projection does not offer this advan- 
tage as in that method, all circles, without exception, 
will appear as ellipses ; consequently, cabinet pro- 
jection has a distinct advantage, and is therefore 
oftener employed when a drawing of an object in 
false perspective is required. 

As an illustration of the principles explained in 
the preceding pages the cabinet projection of the 
pattern for a crank, shown in isometric projection 
in Fig. 215, will be given in Fig. 227. 

At a glance it will be seen that the cabinet pro- 
jection of this object can be drawn in much less 
time than its isometric projection. It is, however, 
necessary to bear in mind, that, whereas all meas- 
urements in isometric projection are equal to the 
actual sizes of the object, those in cabinet projec- 
tion which are parallel to the 45° axis are drawn 
equal to y^ of their actual size. 

Figs. 228, 229, 230, 231, and 232 represent addi- 
tional illustrations of objects drawn in cabinet pro- 
jection. 

Note. — The thorough knowledge of cabinet and isometric projec- 
tions will be of great advantage, both to the student and the mechanic, 
as they will thereby be enabled to represent different objects in drawing 
in such a manner as to be easily understood by persons who would not 
understand a mechanical drawing executed in another, though perhaps 
a more generally approved manner. 



ROGERS' DRAWING AND DESIGN. 



187 




Fia. 229. 




Fig. 230 




Fui. 231. 




Fig. 232. 



128 



ROGERS' DRAWING AND DESIGN. 



ORTHOGRAPHIC PROJECTION. 

Isometric drawing and cabinet projection, while 
showing the object as it really appears to the eye 
of the observer, are neither of them very convenient 
methods to employ where it is necessary to measure 
every part of the drawing for the purpose of repro- 
ducing it in the shop. 

All shop drawings, or working drawings as they 
are usually termed, are made by a method known 
as orthographic projection ; in isometric or cabinet 
projections, three sides of the object are shown in 
one view, while in a drawing made in orthographic 
projection, but one side of the object is shown in a 
single view. 

To illustrate this, a clear pane of glass may be 
placed in front of the object intended to be repre- 
sented. 

In Fig. 233 a cube is shown on a table ; in front 
of it, parallel to one face (the front face) of the 
cube, the pane of glass is placed. 

Now, when the observer looks directly at the 
front of an object from a considerable distance, he 
will see only one side, in this case only the front 
side of the cube. 

The rays of light falling upon the cube are re- 
flected into the eyes of the observer, and in this 
manner he sees the cube. The pane of glass, evi- 



dently, is placed so that the rays of light from the 
object will pass through the glass in straight lines, 
to the eye of the observer. The front side of the 
object, by its outline, may be traced upon the glass, 
and in this manner a figure drawn on it (in this case 




Fig. 2:«. 



a square) which is the view of the object as seen 
from the front. This view is called the front eleva- 
tion. - 

One view, however, is not sufficient to show the 
real form of a solid figure. In a single view two 



ROGERS' DRAWING AND DESIGN. 



129 



dimensions only can be shown, length and height ; 
hence the thickness of an object will have to be 
shown by still another view of it, as the top view. 

Now, place the pane in a horizontal position 
above the cube which is resting on the table, Fig. 




Fig. ast. 



234, and looking at it from above, directly over 
the top face of the cube, trace its outline upon the 
pane ; as a result, a square figure is drawn upon the 
glass, which corresponds to the appearance of the 



cube, as seen from above. This square on the glass 
is the top view of the cube, or its "//aw." 

In Fig. 235 is shown the manner in which a side 
view of the cube may be traced ; the glass is placed 
on the side of the cube, which rests on the table as 
before, and the outline of the cube on the glass in 
this position, is called its ''side elevation.'' 

Usually either two of the above mentioned views 
will suffice to show all dimensions and forms of the 
object. In complicated pieces of machinery, how- 
ever, more views, three and even more may be re- 
quired to adequately represent the proportions and 
form of the different parts. 

A drawing which represents the object as seen 
by an observer looking at it from the right side is 
called the right side elevation and a drawing show- 
ing the object as it appears to the observer looking 
at it from the left side is called the left side eleva- 
tion. 

A view of the object as seen from the rear is 
called the rear view or rear elevation, and a view 
from the bottom, the bottom view. 

The different views of an object are always ar- 
ranged on the drawing in a certain fixed and gener- 
ally adopted manner, thus — 

The front view is placed in the center ; the right 
side view is placed to the right of the front view, 



130 



ROGERS' DRAWING AND DESIGN. 



and the left side view to the left ; the top view is 
placed above the front view and the bottom view 
below it. The different views are placed directly 
opposite each other and are joined by dotted lines 
called projection lities. 




Fig. 23.1. 



By the aid of projection lines, leading from one 
view to the other, measurements of one kind may 
be transmitted from one view to the other ; for in- 
stance, the height of different parts of an object 



may be transmitted from the front view to either 
one of the side views ; in like manner the length of 
different parts of the object may be transmitted by 
the aid of projection lines, to the bottom view and 
top view. 

It is often desirable to show lines belonging to 
an object, although they may not be directly visible. 
In Fig. 236 the top view and the bottom view show 
plainly that the object is hollow ; looking at the 
object from the front or from the sides, however, 
the observer could not see the inside edges of the 
object, except it were made of some transparent 
material. 

For mechanical drawing, we may assume that all 
objects are made of such material, transparent 
enough to show all hidden lines, no matter from 
which side the object is observed. It is the gen- 
eral practice to draw the hidden edges by lines 
made of dashes — dash lines — as in Fig. 236. 

In the following articles the student will find a 
number of exercises on the application of ortho- 
graphic projection. 

Note. — Mechanical drawing is used mainly to represent solids, but 
solids are bounded by surfaces -whicb in turn are bounded by lines 
which by themselves are limited by points ; views of a solid can there- 
fore be found hy drawing the views of its limiting points, lines and 
surfaces, according to the principles of orthographic projection. 



ROGERS' DRAWING AND DESIGN. 



131 



Tof3 Vi 



ew 



Lefl 'SideVitu/ 



t I 



Front View I • HigJii Side Vteu 



JBoitom View 

Fig. 236. 



133 



ROGERS' DRAWING AND DESIGN. 



Draw the front view, left side view and top view 
of the rectangtilar prism showji in .Fig. sjj. 

Fig. 238 shows the drawing of the prism in the 
three required views ; the lines showing the dimen- 
sions are made by long dashes drawn very thin. 




FlO. 237. 



It is important to remember that dimension lines 
must be drawn parallel to the distances, the size of 
which they are intended to show. The dimension 
lines terminate in arrow heads drawn with an ordi- 
nary writing pen. If a dimension line is carried 
outside of a view, short auxiliary dotted lines are 
employed to join the part of the object to which the 
dimension line refers. The dimension line is left 



open near the middle, where the figure denoting 
the measurement is placed. These figures should 
be written very plainly and placed so as to read 
along the dimension line ; for horizontal lines from 









/v 




< 

1 • : 


• 

t 


1, 




^ y" — 


— •? 



i^G. S38. 



the bottom of the drawing, and for vertical lines 
from the right hand side of the drawing. 

The inch is marked " the foot ' — for example : i 
foot, 3 inches is represented by i' 3". More infor- 
mation concerning dimensions will be found in the 
chapter treating on working drawings. 



ROGERS' DRAWING AND DESIGN. 



133 



Draw a front view, top view and right side view 
of the wedge showji in Fig. 2^0. 

Draw the front view first. Lay off a straight 
line, on which mark two points 3" apart ; through 
the point on the right erect a perpendicular, which 
make one inch long ; two sides of the right-angled 




triangle forming the front view of the wedge are 
thus found. 

Join the two ends of these sides by a straight line 
and the front view is complete. The student will 
draw the side view and the top view in correspond- 
ing positions to the right side and above the front 
view, as in Fig. 239. 




Fig. 240. 




Fig. 239. 



134 



ROGERS' DRAWING AND DESIGN. 



Draw a front view, both side views and top view 
of the object shown in Fig. 2^1. 

Fig. 242 shows the required views of the object. 
The edge ab which is visible in the right side view 



will not make the understanding of the view more 
difficult. 

Whenever the view is so complicated that any 
additional lines would only tend to obstruct a clear 
conception of the object, it is advisable to carry the 
dimension lines outside of the view. 

Dimension lines must necessarily be of three 




Fia. 241. 



is hidden in the left side view and therefore is 
represented by the dash line cd. 

It may often be possible to put in the dimension 
lines within the views, when the object is not of a 
complicated nature, and when the dimension lines 



kinds: i, parallel to the lengths of the different 
parts of the object ; 2, parallel to the width of these 
parts, and 3, parallel to the height. The dimension 
line must always be parallel to the line or edge whose 
length it represents. 



ROGERS' DRAWING ANQ DESIGN. 



135 




















; 


V 














d 




--...^ 


, a 






P-- 


^ Jf- 


— --^ 






> 


V 










\ 


t 



Fig. 243. 



136 



ROGERS' DRAWING AND DESIGN. 



Draw a front view, side view and top view of the 
model shown in Fig. 24J. As the object to be 
drawn has the same appearance from either right or 
left side, it does not matter which side view is to be 
drawn. 




Fig. 243. 

The construction of the views is so obvious that 
no explanation need be offered with the drawing 
shown in Fig. 244. It will be noticed that this 
figure, as well as all others in this chapter, are 
shown with lines representing the sides of the dif- 
ferent parts of the object. 



Draw two elevations {a front view and a side 
view^ and a top view of a hexagonal prism f long 
and 2y2" between any tivo parallel sides. 



A 

X"" 

■V. 


























Fig. ;244. 

It is evident that a hexagonal prism has six faces 
and of these three are parallel to the remaining 
three faces. The distance between any two parallel 
faces or sides is the same ; in this case it is equal 
to 23^ "; let us draw the top view of the prism first 
of all. 



ROGERS' DRAWING AND DESIGN. 



137 



N 



Draw two lines, AB horizontal and CD vertical, Fig. 245, intersecting 
each other at the point O. If it is intended, as in this case, that the 
intersection, O, of these two lines should coincide with the center of the 
view which is to be drawn, then these lines are called center lines ; the 
use of center lines in projection drawing is very extensive. 

Make the line CO equal to OD and each equal to 1%", so that the 
line CD is equal to 2j4", the distance between the parallel sides of the 
prism. Through C and D draw the lines eCd and aDb parallel to AB ; 
then through the point O draw two 60-degree lines eb and ad, cutting 
the lines eCd and aDb at the points e, b, 
a and d. 

Through these points draw the re- 
maining sides of the hexagon, parallel to 
the lines eb and ad. The hexagon, 
aAedBb shows the top view of the prism. 

To draw the front view proceed as fol- 
lows : 

Through the points Aab and B draw 
the vertical lines AE, aH, bj and BF. 

Draw the horizontal line NGP, make 
PF equal to 5", the height of the prism 
and through the point F draw the hori- 
zontal line KEF ; then the figure, EHJ 
FPG is the front view of the prism. 

It will be noticed that the front view 
shows three faces of the prism : HEGS, 
HSRJ and JRPF, the faces HEGS and 



f 




^ 



K 



M 



Fig. 245. E 



H 



138 



ROGERS' DRAWING AND DESIGN. 



JRPF appear narrower than the face HSRJ, the latter being situated 
right in front of the observer and parallel to the plane of the paper is 
seen in its true size, while the other two faces seen in the front view 
being in an inclined position relative to the front face appear narrower 
than their true width. 

The side view KNTLM shows only two faces of the prism. The dis- 
tance KM is equal to CD, the edge LT corresponding to the edge 
marked by the letter A in the top view, cuts the line KM into two equal 
parts, KL and LM. 



To draw the top view, front view and side view of 
a hexagonal pyramid 5" high, each side of the hex- 
agonal base being equal to i%". The top view of 
the pyramid must be drawn first. 

Fig. 246 shows the required views. The top 
view appears as a regular hexagon, in which all 
diagonals are drawn by lines as heavy as the sides, 
as these diagonals show the edges of the faces of 
the pyramid. The center of the hexagon where all 
the diagonals meet represents the vertex of the 
pyramid. 

The front view and the side view are drawn in 
the manner explained in the construction of these 
views of the hexagonal prism, the edges of the faces 
in this case all meeting in the vertex which is 
placed 5" above the middle of the line representing 
the base. 




ROGERS' DRAWING AND DESIGN. 



139 



To draw a top view ajid a front view of an octag- 
onal prism. Let each side of the octagonal bases 
be equal to one inch and let the height of the prism 
be 8". 



To complete the front view, intersect these lines by 
two horizontal lines 8" apart. The side view of this 
figure is identical with the front view. 







Draw the top view first. Fig. 247 shows the re- 
quired views. The top view is an octagon, each 
side of which is equal to one inch. The front view 
is drawn by projecting vertical lines from the points 
a, b, c, and d of the octagon. These vertical lines 
form the vertical edges of the faces of the prism. 



Fig. 248 shows three views of a sphere, each of 
which appears as a circle. 

The lines, AB, CD, EF and GH are center lines. 
They are composed of long and short dashes, alter- 
nating, and are usually extended indefinitely beyond 
the outlines of the views. 



140 



ROGERS' DRAWING AND DESIGN. 



Center lines are drawn through the middle of the 
view in all cases where such a line will divide the 
view into two perfectly equal parts so that one part 
will have all its details situated opposite the corre- 
sponding details of the other part, so that if the 
paper on which the view is drawn is folded along 
the center line, all parts in one half of the view will 
cover exactly all corresponding parts in the other 
half of the view. 

We say then that the view (or object) is sym- 
metrical with regard to the center line. In Fig. 245 
the top view and the front view are symmetrical 
with respect to the center line CD. 

The top view, however, may be folded along the 
line AB, and in this case the lines of the hexagon 
on one side of the line AB will exactly cover the 
lines in the other half of the hexagon ; we see then, 
that the hexagon is symmetrical in regard to the 
center line AB also. 

In all cases where a view is symmetrical in re- 
spect to two lines, both of these lines must be 
drawn. Wherever the view is symmetrical to one 
line only, not more than one center line must be 
drawn ; in Fig. 248 all views are symmetrical to 
both horizontal and vertical center lines ; center 



lines continued from one view to the other show 
that the views belong together, just as projection 
lines would indicate the same. 




B 



Pig. 248. 



A center line should never be used as a dimension 
line, but such lines may be laid off from the center 
line on both sides of it. 



ROGERS' DRAWING AND DESIGN. 



141 




Fro. 249. 



In Fig. 2^g is shown the top view (or plan) and 
front view {or elevation) of a cylinder, j" high and 
i%" in diameter. The top view is a circle ij^" in 
diameter, the front view a rectangle 3" by i^". All 
side views of the cylinder are the same. 

As in all figures standing on a base of an irregu- 
lar shape, the top should be drawn in this case be- 
fore the front. The width of the front view is 
determined by projection lines from the top view ; 
observe that the top view has two center lines, a 
horizontal and vertical one ; the front view has 
only one line of symmetry, the vertical. 

Draw the front view and side view of a cylindrical 
pipe 8" long^ outside diameter f, inside diameter j" ; 
in Fig. 250 the required views are shown. 

The two dash lines in the front view show the 
inside walls of the pipe, which are represented in 
the top view by the smaller circle. 

Fig. 250 may also represent two views of a pipe 
into which a cylinder has been inserted. We have 
here an interestinof illustration of a case where two 
views of an object, a front view and a top view, do 
not define sufficiently the true character of the 
object represented. A similar difificulty may arise 
with most hollow objects, and it is evident that 
some method must be devised to overcome any 




Fig. 230. 



142 



ROGERS" DRAWING AND DESIGN. 



such misunderstanding as to the true nature of the 
object represented. 

This may be done by representing the front view 
of the pipe as if it were cut in half like the cylinder 




Fio. 2r,i. 

shown in Fig. 251 ; a front view of such a pipe cut 
in half is shown in Fig. 252 ; the top view is that of 
a whole pipe. The line 1-2 shows the manner in 
which the cylinder is supposed to be cut, and is 
called the line or plane of section. 

The front view in Fig. 252 we call the section 
view or section on 1-2. The line of section should 
be mads up of dashes alternating with two dots. 




cSection on /-%. 

Fio. 252. 



ROGERS' DRAWING AND DESIGN. 



143 




The inner part of the material of the pipe ex- 
posed by cutting, is covered by lines about xVth 

I 




inch apart and inclined 45 degrees. Fig. 253 shows 
the same pipe with only a portion of its upper half 
cut away; in Fig. 255 is shown this partial section 
of the pipe. 




Fig. 2.56. 



Fig. au. 



144 



ROGERS' DRAWING AND DESIGN, 




In Fig. 254 is shown still another way of cutting 
the pipe, and in Fig. 256 appears the corresponding 
front view, with a similar partial section. 



Within the pipe de- 
scribed in the preceding 
problem {8" long, ^" outside 
and j" inside diameter^ is 
placed another pipe 8" 
long, j" outside diameter 
and 2" inside diameter. 

Draw the top view and 
section of these two 
pipes. The top view 
(Fig. 257) shows three cir- 
cles, 4', 3" and 2" in diam- 
eter ; the section on the 
line AB shows one-half 
of one pipe within the half 
of the other pipe. The 
section lines in the one 
pipe run in a different 
direction from those in 
the other ; this is done in 
order to show more dis- 
tinctly that there are two 
separate pipes. 



I 



i 



m 






Fig. 257. 



Draw two views of a cylindrical ring. 




Fig. 258. 



Fig. 258 shows the plan and section of such a ring. 
The drawing does not require any special explanation. 



ROGERS' DRAWING AND DESIGN. 



Draw two views of the cylinder with square Jlange 
shown in Fig. 2^g. 

Let the side of the cylinder be lo" long (entire 
leno-th) outside diameter 4", inside diameter 3", 
and the flange 6" by 6" and ^'2" thick. 

The flange has four bolt holes, each i/^" in diam- 
eter. 

The top view and section of this figure are shown 
60. 



in Fig, 




Fig. 259. 




146 



ROGERS' DRAWING AND DESIGN. 



Fig. 261 shows the top view and two sections of a 
bed plate. 



view, is a section parallel to the short side of the bed 
plate and is called a cross section or a lateral section. 




Jiongitudina? <Seci'ion on JiS. 



V/VXA/^//-. 



^22\ 



Y^zz 



6 



Fio. 2(il. 



The longer section which shows the appearance 
of the bed plate when cut in a plane parallel to the 
longest side of it, is called the longitudinal section. 
The other section, placed to the right side of the top 



In Fig. 262 is shown a top and front view, and 
lateral section of a hexagonal nut. 

The figure shows an arrangement of views which 
is frequently adopted in order to economize space. 



ROGERS' DRAWING AND DESIGN. 



147 



Fig. 261 is an illustration of the same principle. 
In that figure the advantage of this method of ar- 
ranofement is even more striking • the breadth beine 
considerable, as compared to the height, it is evi- 
dent that, if the lateral section had been placed in 
line with the longitudinal section, the three would 
have occupied more space than with the arrange- 
ment shown. 

In explaining the way in which we derive the 
different views of an object, we have placed it in 
Figs. 233, 234, 235 on a table, and in front of the 
object (a cube in this case) we hold a pane of glass 
and in all the illustrations, we have placed the pane 
of glass parallel to one of the faces of the cube. 

In the exercises in projection so far taken up, we 
have placed the object in a similar position ; that is, 
one of the faces of the object was supposed to be 
parallel to the table or paper on which the drawing 
was to be made, the sides of the object were either 
horizontal or vertical and the center lines were also 
either horizontal or vertical ; it is always desirable 
to select such a position for the object which is to 
be drawn. 

Small parts of machinery, shown in detail draw- 
ings, are nearly always drawn in this manner. It 
may happen, however that some parts of a machine 
will appear with their sides at different angles to 



the plane of the paper, or it may even be desirable 
to place the object in such a position purposely. 

Drawings made in this manner, will as a rule, 
offer more difficulties to the draughtsman, as most 




Fig. :X2. 



views will appear more complicated when the ob- 
ject is placed in an inclined position. The follow- 
ing exercises will show the objects drawn before, 
now placed at different angles. 



148 



ROGERS' DRAWING AND DESIGN. 




long and i 

forms an angle of 30 degrees 



Fig. 263. 



Drazc a front view, top view and right side view 
of a prism ^"x ^"x /" standiiig with its face ^"x /" on 
a horizontal plane, and the long vertical side of the 
prism forming an angle of jo degrees with the lower 
edge of the drawing board. 



Draw the top view (plan) first ; to 

do this, draw the rectangle ABCD 4" 

wide, so that the line CD 

with a 

horizontal line, Fig. 263. 

To draw the front view, draw the 
horizontal lines, EH and KN 2" apart ; 
from the points ABCD in the top view 
draw vertical lines cutting the lines EH 
and KN at the points E, K, F,and L, 
G and M and H and N. 

The figure KEFGHNML is the 
front view, FL being the vertical edge of the prism 
nearest to the observer, designated in the plan by C, 
and the most distant (hidden) edge, corresponding 
to the point B in the plan is MG shown in dash 
lines. 

The side view is placed opposite to the top view 
in this case, as in this position the construction of it 
is much easier. 

The, vertical edges of the prism will appear hori- 
zontal in this position of the side view. They are 
drawn from the points ABCD in the top view 
and the edges TRO and PSU, being the lower 
and the upper faces of the prism, are two inches 
apart. . 



ROGERS' DRAWING AND DESIGN. 



149 



Draw a front view, side view and top view of the 
prism described in the last exercise, placed so that the 
face forming the base of the prism, f'si. i" is inclined 
45 degrees to the paper and the front face, fx 2" re- 
mains vertical to 

the paper and par- • ' 

allel to the lower 
edge of the drazv- 
ing board. Fig. 
264. 



Draw the front view first ; it is a rectangle 4". long 
and 2" wide, the long side of which forms an angle 
of 45 degrees with a horizontal line. 

The figure shows plainly how the top view may 
be constructed by projecting vertical lines 
from the front view. In the same manner 
the side view may be drawn, when it is 
placed opposite the front view, as in Fig. 
264. 




Fig. 264. 



150 riOGERS' DRAWING AND DESIGN. 








Draw the wedge shown in Fig. 26 £, placed so that 
the sides of the square corner of it form angles of 4^ 
degrees with the plane of the paper, with both tri- 
angular faces vertical to the paper and parallel to 
the lower edge of the drawing board. 

The front view showing the triangular face is 
drawn first. The two sides of the triangle, which 
form the right angle (the legs) are drawn at- 45 
degree angles to a horizontal line. 

The construction of the top view and side view is 
plainly shown in Fig. 265 and requires no special 
explanation, as they are drawn in the same manner 
described in the drawing of the top and side views 
in the two preceding exercises. 












-\ 




^-^^ 1 










W \j/ 






L__ 


FiQ. 265. 





ROGERS' DRAWING AND DESIGN 




The object shown in Fig. 266 is placed with its 
base upon a horizontal plane (the plane of our draw- 
ing) while the two vertical faces visible to the observ- 
er, are placed respectively at angles of 30 degrees and 
t>o degrees to the lower edge of the drawing board. 

Let it be required to draw the front view and top 
view of tliis object. 

Draw the top view first. 

To do this, draw the rectangle A BCD, AD par- 
allel and equal to BC, 4" long, and DC equal and 
parallel to AB, 2" long ; AD is inclined 30° and 
DC 60° to a horizontal line. Fig. 266. 

To draw the front view, draw the horizontal line 
FM, and through the points A, B, C and D draw 
vertical lines meeting the line FM at the points 
F, H, K and M. 

On the side AF set off the distance FE equal to 
3"; through the point E draw the horizontal line 
EG cutting the line BH at G. 

On the line MC set off the distance MN equal to 
2" and through the point N draw the horizontal 
line NL, cutting the vertical line DK at L, join the 
points E and L, G and N. This completes the 
required front view; EF and GH are the two 
longer vertical edges of the object, GH being 
hidden. LK and MN are the two shorter vertical 
edges of the object, both visible. 



152 



ROGERS' DRAWING AND DESIGN. 




In Fig. 267 is shown the front 
view, top view and side view of 
the object just drawn. 

In this instance the object is 
placed with its longer vertical 
edge nearer to the observer ; 
otherwise the position of the 
object is exactly the same as 
described in the preceding exer- 
cise. _/ 

The side view is drawn in a 
manner similar to the front view, 
by lines projected from alt points 
(corners) of the top view. 

No doubt the student has noticed that in drawing an 
object placed at an angle to the lower edge of the drawing 
board, but having two faces parallel to the plane of the 
paper, we draw the top view first ; that is, the view of it, 
which being parallel to the board, will appear in its simplest 
outline, with all lines drawn in their true length and posi- 
tion. 



ROGERS' DRAW1NC3 AND DESIGN. 



153 




Fig. 288. 



To draiu the front view and top vieiu of a hex- 
agonal prism, standing upon a horizontal plane and 
having two of its parallel vertical sides, parallel to 
the lower edge of the drawing. 

Let each side of the hexagon forming the bases 
of the prism be equal to one inch and the height of 
the prism be 4"; the top view is drawn first ; it is a 
regular hexagon, length of each side being i", Fig. 
268. 

The front view is drawn by projecting lines from 
the corners of the hexagon shown in the top view, 
these lines making the vertical edges of the prism, 
and then intersecting these lines by two horizontal 
lines 4" apart, thus forming the top and the bottom 
of the prism. 

If an object is to be drawn, placed so that it is 
inclined to the plane of the paper, but having its 
front face parallel to the lower edge of the drawing 
board, the front view is drawn first. 

As a rule it will be observed that that view is 
draivn first, which is drawn easiest, and especially 
the view which shows the object in its true form ; 
the other views are drawn by projection from the 
different points of the view completed. 



ROGERS' DRAWING AND DESIGN. 




Let it be 7^equired to dratv the 
top view and front view of the 
same prism as in the same exer- 
cise, but placed so that tivo of its 
parallel vertical sides are paral- 
lel to the lower edge of the draw- 
ing board and t lie base inclined to 
the plane of the paper at an angle 
of 30". 

Draw the front view agd top 
view of the prism, Fig. 269, 
showing the prism standing in a 
vertical position ; WVZY is this 
front view and TORSU is the 
corresponding top view. 

To draw the front view of this 
hexagonal prism with its base 
inclined at an angle of 30 de- 
grees, draw a line AF making 
an angle of 30 degrees with a 
horizontal line. 

Upon this line erect the rect- 
angle which is exactly equal to 
the front view of the hexagon 
in its vertical position, as shown 
in the same figure by WVZY. 



ROGERS' DRAWING AND DESIGN. 



155 



To draw the top view, extend the 
horizontal lines of the top view RS, 
TU indefinitely ; then draw vertical 
lines through the points B, C, D and 
E; tliese lines intersect the horizon- 
tal lines RSK and TULO and the 
center line OGNP in the points, 
LGJINM forming the upper face of 
the prism in the required top view. 

To complete the top view draw 
vertical lines through H and F cut- 
ting the line JK at the point K, the 
line GP at the point P and the line 
LO at the point O. 



To draw a hexagonal pyramid, 
having the sizes of the hexagonal 
prism in the preceding exercise, and 
placed in the same position. 

The construction in this case is 
exactly the same as in the last exer- 
cise ; Fig. 270 shows the required 
drawing. 




Fig. 270. 



156 



ROGERS' DRAWING AND DESIGN. 



S E 




ROGERS' DRAWING AND DESIGN. 



157 



To draiv the fop z>ic-a' and fro)it vieio of a cylinder 
7^'/iose axis makes an angle of 60° zvit/i a horizontal 
line and u</iic/i lies in a vertical plane parallel to t/ic 
loiver edge of the board. 

In Fig. 271 ABCD is the front view and EF the 
top view of the cylinder, when its axis is vertical. 

Draw the line 3'h at 60° to the horizontal and let 
this be the center line of the cylinder in the re- 
quired inclined position, corresponding to the 
center line in the front view of this cylinder in its 
vertical position. 

Make abed equal to ABCD for the required 
front view. The top view is drawn in the followino- 
manner : divide the horizontal diameter KL into 
any number of equal parts, say six. Through the 
points of division, i, 2, 3, 4 and 5 draw vertical 
lines MN, OP, EF, RS and UT. 

Divide the line dc into the same number of equal 
parts, marked by the division points i', 2', 3', etc., 
and draw vertical lines through all these points, as 
well as through the points c and d. 



The vertical lines passing through the points d 
and c cut the horizontal center line of the top view 
in the points 1 and k. The vertical lines drawn 
through the points i', 2', 3', etc., cut the center line 
Ik in the points i", 2", 3", etc. 

Through the points E and F draw two horizon- 
tal lines cutting the line 3"3' in the points e and f. 

Through the points IVI and N draw two hori- 
zontal lines intersecting the line i"i' at the points 
u and t and cutting the line 5 "5' at the points m 
and n. 

Through the points O and P draw two horizontal 
lines cutting the vertical lines 2"2' and \ /( at the 
points s, r, o and p. 

A curve traced through the points I. u, s, e, p, n, 
k, m, o, f, r, t will be the required projection of the 
upper base of the cylinder. 

The lower base may be constructed in exactly 
the same manner. 



158 


ROGERS' DRAWING AND DESIGN. 




^^.^<7 ^\ , To draiv the section of a cylinder made by an tn- 




!5c ^\ >? dined filane. 
X "■- ^ --' \ 




V ^'^N ^^"\-'''' V- 








/\ ""v ^-'■■' \ ^v 




/ ^ X f \ \ w 




/ X \ y \ ^» 




/ \ '^ - ' . 




«/ ^. ^'^< ^ 




y \- '^' ^-x /\ \ 




\ ' \ >■ / ^ ". 




\ - / \ \ / \ \ \ / \ 




\ ^ X V / '' ^. > N 








V- — ^-J<-^ _— ^^— 1 — -^- ^T -X v. \» " 


\ s2l i.X.^>.-.-. X :::/ 












/ V V 1 * \ ■* ^ /^ 






f-x --M^---v-Y-v 7< 






■«/ \ i \ \ \ ^ ^\^ y^ 












\ ^ i ^^ / ^^^ y' 












\ 1 *^^ ^v'^ X 






\. 1 \ ^/K^ 






**^\" T^~"\" 'yTy^'^ 






\^^_j__,><x 




"^ * Fig. 273. Jb 





ROGERS' DRAWING AND DESIGN. 



159 



The front view of such a cylinder is shown in 
Fig. 2"] 2 by abc6', where the line c6' shows the in- 
clination of the plane which makes the section. 
Above the front view is shown the top view of the 
cylinder as the circle dgmne. 

To draiv the form of the section, divide the hoi'i- 
zontal diameter d6 into any number of equal parts. 

In Fig. 272 this diameter is divided into 6 equal 
parts. 

Through each one of the division points thus ob- 
tained, draw vertical lines, which intersect the circle 
at the points f and e, g and h, i and j, k and 1, m 
and n ; the same vertical lines will cut the inclined 
line c6' at the points i', 2', 3', etc., which points will 
divide the line c6' into six equal parts. 

Through the points c, i', 2', 3', 4', 5', 6' draw lines 
perpendicular to the line c6'; at any convenient dis- 
tance from this line draw the line 6"A parallel to it, 
and this line will be cut by the perpendiculars at 
the points i", 2", 3", 4", 5", thus being divided into 
six equal parts the same as the parts of the line c6". 



Now set off Bi" equal to i"C each equal to if; 
the same distances set off from the point 5", making 
KL equal to BC. 

Lay off 2"E equal to 2"D and each equal to 2g, 
and the same distances set off from the point 4", 
making JH equal to ED ; then make 3"G equal to 
3"F and each equal to 3i so that ii is equal to FG. 
This line FG is the minor axis and the line A6" is 
the major axis of an ellipse, which may be traced 
through the points ACEGJL6"KHFD and B and 
which forms the required section. 

It will be noticed that the section of a cylinder 
made by a plane which does not intersect any of the 
bases of the cylinder, and which is not parallel to 
the bases (that is, perpendicular to the center line), 
is an ellipse ; when the cutting plane is parallel to 
the bases the section produced is a circle, just like 
the bases ; when the cutting plane passes through 
the center line the section is a rectangle, two oppo- 
site sides of which are equal each to the height of 
the cylinder. 



160 



ROGERS' DRAWING AND DESIGN. 



An ellipse may be produced also by cutting a 
cone by a plane which does not intersect the base 
of the cone, as in Fig. 273, where the line ab indi- 
cates the cutting plane. 




Fig. 273. 



When the cutting plane, ab, Fig. 274, is parallel 
to the line cd, then the figure produced in section 
is a curve known as the parabola. Such a curve is 
shown in Fig. 275. 




Fia. 275. 




ROGERS' DRAWING AND DESIGN. 



161 



Any point, P, of the parabola is equally distant 
from the line DD and from the focus F, so that 
FP is equal to PA. 

When the cutting plane intersects the base of the 
cone and is not parallel to any one of the lines on 
its surface, the curve produced in a section is called 
a hyperbola. In Fig. 276 ab indicates the cutting 
plane. A hyperbola is shown in Fig. 277. 




Fia. 2T7. 

The distance between the distance PF' of any 
point in the hyperbola from one of the foci, and the 
distance, PF, the distance of this point to the other 
focus, must always be equal to a given line. 

In the chapter on geometrical drawing several 
methods have been explained in which an ellipse 
may be drawn ; the way to draw a hyperbola has 
also been shown. Of these curves the ellipse is 
ofttimes encountered in mechanical drawing. 



163 



ROGERS' DRAWING AND DESIGN, 





Fig. 278. 



ABCDEFGHIH'G 

Figs. 279 and 280. 



F' E' D' C B' A 



ROGERS' DRAWING AND DESIGN. 



163 



DEVELOPMENT OF SURFACES. 

The principles of projection just explained may 
be readily applied to the very important problem 
of development of surfaces. 

Whenever it is necessary to make an object of 
some thin material like sheet metal, as in the case 
of boiler connections, etc., the surface of the desired 
object is laid out on sheet metal, in one or in several 
pieces ; these are called the patterns of the object ; 
the pattern being first laid out on the sheet metal 
and then cut out ; when this is done the separate 
pieces are ready to be fitted together to form the 
required object. 

The method by which the surface of an object is 
laid out on a plane is called the development of the 
object. A few exercises will sufficiently acquaint the 
student with the methods used in problems of this 
character. 

To draw the development of a right elboiv, Fig. 
^78. 

A right elbow is made by joining two pieces of 
pipe for the purpose of forming a right angle. It 
is reall)- an intersection of two cylinders of equal di- 
ameters ; the center lines of the two cylinders meet- 
ing at one point, and as the joint is to be a right 
elbow, the center lines must be perpendicular to 
each other. 



Now, to draw the development of a cylinder, pro- 
ceed as follows : 

Divide the circumference of the cylinder into any 
number of equal parts, and through the points of 
division draw lines parallel to the center line of the 
cylinder. 

On these parallel lines mark the points which be- 
long to the curve of intersection with another cyl- 
inder, or any other figure as happens to be the case, 
and then roll out the surface of the cylinder into a 
flat plate. The rolled-out surface will be equal in 
length to the circumference of the cylinder, and it 
will contain all parallel lines, which were drawn upon 
the cylinder, with spaces between them just equal 
to the actual space between the parallel lines which 
were drawn upon the surface of the cylinder. 

By marking the points of intersection on the par- 
allel lines in the rolled surface, the development of 
the cylinder or its part is obtained. In Fig. 278 
the circle showing the circumference of the pipe is 
divided into any number of equal parts by the divi- 
sions I, 2, 3, etc. Lines are drawn through these 
divisions parallel to the center line of the vertical 
portion of the joint. These lines are ak, bl, cm, 
dn, etc. 

The points k, 1, m, n, o are the points on the par- 
allel lines designating the curve of intersection. 



164 



ROGERS' DRAWING AND DESIGN. 



The development of the two branches of the right 
elbow are shown in Figs. 279 and 280; the length 
of the development, VW (or AA') is equal to the 
circumference of the figure shown in Fig. 278. To 
obtain this length all spaces, i, 2, 3, 4, etc., laid out 
upon the circle in Fig. 278 are set off upon a straight 
line ; these spaces are marked in Fig. 2S0 by A, B, 
C, etc., perpendiculars AK, BL, CM, etc., are drawn 
through the points A, B, C, etc. The perpendicular 
AK and K'A' in Fig. 280 are each equal to ak in 
Fig. 278. The second lines on each side of the 
development, the lines BL and B'L' are equal to bl, 
Fig. 278. 

The third lines on each side of the development, 
the lines CM and CM' are equal to the third line 
cm, Fig. 278. 

The fourth lines in the development are made 
equal to the fourth parallel in the elevation. Fig. 
278, and in the same manner all other lines in the 
development are made equal to the corresponding 
parallels in the elevation of the pipe in Fig. 278. 

The middle line, SI in the development is made 
equal to the line si in the elevation ; the points 
KLMSM'L'K', etc., thus found, define the position 
of the curve of intersection in the development 
of the cylinder. 



The required curve is traced through these 
points ; the development AA'K'K is the pattern for 
the part aksi of the right elbow shown in Fig. 278. 

The other part of the elbow is developed in Fig. 
279. It will be readily seen that the figure TVWU 
is laid out in the manner in which the first develop- 
ment was obtained ; in this figure the shortest par- 
allels are laid off above the longest parallels in the 
first development. This arrangement gives the ad- 
vantage of cutting out both branches of the right 
elbow from one square piece of sheet metal wij;hout 
any waste of material. 

It will be noticed that the patterns shown in Figs. 
279 and 280 do not provide for the lap by which 
the two branches are held together. A lap of any 
desired width may be added to the pattern, after it 
is constructed by drawing" an additional curve, par- 
allel to the curve of the above pattern, the distance 
between the two curves being equal to the width of 
the desired lap. 



ROGERS' DRAWING AND DESIGN. 



165 



To draw the pattern of a tee-pipe in which all 
branches are of cqtial diameter. 

In Fie. 281 is shown the front view and the side 
view of a tee-pipe. It is made by the intersection 
of two cylinders of equal diameters; the section of 



The greater the number of these divisions the 
more accurate will be the resultant pattern. 

Through the divisions i, 2, 3, etc., draw horizon- 
tal lines cutting the horizontal cylinder in the side 
view in the points i"i', 2"2', ^i' t>\ 4 4', 5"5', 6"6', 7"/' ; 




Fid. 2H1. 



the cylinders is represented in the front view by 
two 45-degree lines, ad and dg. 

To develop the pipes divide the circle in the end 
view, Fig. 282, into any number of equal parts, in 
this case let it be twelve parts. 



the line 4"4' just meets the lines of the section in 
the point d. The line 5"5' cuts the lines of the 
section in the points e and c, the line 6"6' cuts the 
section lines in the points f and b and the line ""7' 
cuts the lines of the section in the points g and a. 



166 



ROGERS' DRAWING AND DESIGN. 



Draw vertical lines through the points a, b, c, d, 
e, f and g. After all these lines are drawn we have 
all that is necessary to complete the development 
of the cylindrical surfaces. 



the opening, into which the vertical cylinder will 
fit. 

The rectangle ABCD has one side AB equal to 
the length of the horizontal cylinder, Fig. 282 ; the 
I 1 







^^ 




1 ^---.^^ 


2- 












2 






/^ 


ia\ 


3" 
4 






d 






3 

4 




4 


ii\ 

C- -H 


5" 




c/ 


;k 


N. e 

1; 




5' 




-\^ 


\6 


trr^J. 


9/ 
/\ 

1 


6^ 


.... b/ 


!J 


>; 




6' 






/I 










i ^"^ 


T 


a 

h 


1 i 


m| 


g 

n 


T 


! 





Fig. 282. 



Fig. 283 shows the development of the horizontal 
cylinder ; the rectangle ABCD is equal to the cyl- 
inder surface. The curve ODGL is cut out within 
the rectangle for the joint which is the outline of 



other side AD is equal to the circumference of the 
circle, showing the end view of the horizontal pipe, 
Fig. 282. The twelve divisions marked on the cir- 
cle are set off on the straight line AD (Fig. 283) so 



ROGERS' DRAWING AND DESIGN. 



167 



cl 




et 



^^ 



ot 



n 



Fig. 283. 



168 



ROGERS' DRAWING AND DESIGN. 



that together they are equal to the circumference of 
the circle. 

The outline of the opening for the intersection 
of the horizontal pipe with the vertical branch is 
laid out in the middle of the rectangle ABCD in 
the following manner : On the middle line 6'6 are 
set off the distances 6'0 and 6G each equal to g;' 



There still remains to be drawn the development 
of the vertical branch of the tee-pipe ; this is found 
in the same manner as the horizontal part, i. e., by 
laying out the surface of the vertical cylinder ; that 
is, by making it equal in length to the circumfer- 
ence of the circle showing the end view of the 
cylinder. The development is shown in Fig. 284. 



l[ 


aj 


3 


4| 


5| 


6l 


7| 


8 


9 


to 




1 


12 

J 


c ~-^ 










'^ J "~^ 


^1 












""E* 


D\ 


E\ 


. 


/g 

y ry 


/n 


Fig. -^4. 


K\ 


l" 


m\ 




N 


X^ 





(or a;") in Fig. 282, on the lines 5'5 and ;'; are set 
off the distances 5'P, 5F, 7'N and 7H each equal to 
the distance 6'f, Fig. 282 (or b6"). The distances 
4'R, 4E, 8K and 8'M are set of? on the lines 8'8 and 
44, to equal the distance es' (or C5") of Fig. 282. 
The lines 3'3 and 99 are touched by the curve of 
intersection in their center at points D and L. 



On the line AB are set off the twelve parts of the 
circumference and in each one of these divisions is 
erected a perpendicular to the line AB; on these 
perpendiculars are laid off successively the length 
of the vertical lines drawn on the surface of the 
vertical branch ; the lines AC, iD, 2E, 3F, G4, 5H 
and 6J in Fig. 284, are equal correspondingly to 



ROGERS' DRAWING AND DESIGN. 



169 



the lines ah, bi, cj, dk, el, fm and gn in Fig. 282. 

Thus one-half of the development ACJ6 is con- 
structed ; the other 6JP12 is exactly equal to the 
first part. 

The method employed in these cases may be ap- 
plied to nearly all developments of cylindrical sur- 
faces ; it consists in drawing on the surface of the 
cylinder, which is to be developed, any number 
of equidistant parallel lines. The cylindrical sur- 
face is then developed and all parallel lines drawn 
in it. By setting off the exact lengths of the parallel 
lines a number of points are obtained, through loliich 
may be traced the outline of the desired development. 

It has been noted in Fig. 282 that the intersec- 
tion of two cylinders of equal diameters — their arcs 
intersecting each other — will always appear in the 
side view as straight lines at right angles to each 
other. If one cylinder is of a smaller diameter than 
the other then the intersection will be a curve. 

Now, let it be required to find the intersection of 
two cxUndrical surfaces when the smaller cylinder 
passes through the larger one, their axes intersecting 
each other. The front view, top view and end view 
of such two intersecting cylinders is shown in 
Fig. 285. 

The larger cylinder is marked by the letter B in 
all views, the smaller one bv the letter A. 



Divide the circle in the top view into any number 
of equal parts, say twelve, marking the divisions of 
tlie numbers i, 2, 3, 4, etc. Through these points 
draw vertical lines cutting the small cylinder in the 
side view in the points t, u, v, w, x, y and z. 

Divisions exactly like those made by these points 
will be set off now on the small cylinder in the end 
view by the points a, b, c, d, etc., through which 
vertical lines are drawn cutting the larger cylinder in 
the points e, f, g, h, etc. Extend the line u 12 down- 
Ward until it meets a horizontal line drawn through 
g. These lines give the point k at their intersection. 

The point 1 is obtained by drawing the line v 1 1 
downward until it intersects with a horizontal line 
drawn through the point f ; the point m is obtained 
by cutting the line 10 w, extended downward, by a 
horizontal line drawn through the point e. 

Thus one-half of the required line of intersection, 
as it appears m the side view, indicated by the 
points s, k, I, m, is obtained, the other half, m, n, o, 
p is exactly the same as the first and may be drawn 
in a similar way. 

Note. — Such a line of intersection is one which is frequently 
encountered in mechanical drawing and it is advisable to retain a good 
idea of its form. In drawing joints or intersections of this kind, it will 
not be required, as a rule, to lay out the section in the above accurate 
manner. Keeping in mind the true section of two cylindrical surfaces 
of different diameters, the student should be ready to sketch the 
required section freehand, approximately true to suffice for practical 
purposes. This is done first in pencil and then in ink. 



170 



ROGERS' DRAWING AND DESIGN. 



The intersection in this 
case, as well as in similar cases, 
where the precise form is 
required, should be traced 
through the obtained points 
carefully and then inked in 
with the aid of an irregular 
curve ; for ordinary purposes 
the section may be represented 
by an arc of a circle, some- 
what approaching the curve 
laid out in pencil. 

The development of the 
upper branch of the smaller 
cylinder is shown in Fig. 287. 
The line AC contains twelve 
equal divisions, each equal to 
one division in the circle in 
the top view, Fig. 285 ; the 
length of the line AC is there- 
fore equal to the circumference 
of the circle, which represents 
the top view of the smaller 
cylinder. 

Through these divisions on 
the line AC are drawn perpen- 
diculars which are made suc- 




Fiu. 285. 



ROGERS' DRAWING AND DESIGN. 



171 



cessively equal to the lines st, ku, Iv, mw, nx, oy, 
and pz ; in this manner one-half, ADEB of the 
desired development of the smaller cylinder is 



the rectangle VWUT, VU being equal to the length 
of the cylinder and VW to its circumference. It may 
be divided into two equal parts, one for the upper 





Fig. 286. 



Ub 



O b b] 



obtained ; the second half BEFC of the develop- 
ment is an exact duplicate of the first half. 

The development of the larger cylinder is shown 
in Fig. 286. The surface of this is represented by 



half of the cylinder, containing one opening for the 
upper branch of the smaller pipe, and the otherhalf 
an exact duplicate of the first one, containing an 
opening for the lower branch of the smaller cylinder. 



172 



ROGERS' DRAWING AND DESIGN. 



To find ihe outline of the opening, draw the cen- 
ter line C M' M for both halves, and the line HD 
at right angles to CM in the middle of the first half 
of the cylinder. 

On the line VW below the point H set off the 
distances HG, GF" and FE equal to the distances 
hg, gf, and fe in Fig. 285; the same distances are 



The other half of the opening is exactly the same 
as the first, and is laid off in the same way on the 
other side of the line M'M. The curve MLKSK' 
L'M PRDQO is the complete opening for one 
branch of the smaller cylinder. In the other half 
of the same figure, a similar opening NN' is laid 
out for the other branch of the smaller pipe. 




laid off above the point H, on the same line, so that 
HG'=HG, G'F=GF and F'E'=FE. 

Draw lines parallel to HD through the point E, F, 
G, G', F', and E' and on these lines set forth the 
distance H S equal to h's (in Fig. 285) GK and 
G'K' each equal to g'k in Fig. 285 ; FL and F'L' 
each equal to f'l in Fig. 285 and the distances EM 
and E'M' each equal to e'm in Fig. 285. Thus one- 
half, MLKSK'L'M' of the opening is obtained. 



The rectangular piece, VWTU, with the two 
openings. MM' and NN', is the required pattern of 
the larger cylinder. 

To draiu the development of a four-part elbow. 

A four-part elbow is a pipe joint made up of four 
parts, such as is used for stove-pipes ; in Fig. 288, 
the four parts forming the elbow are AKSI, 
KXTS, XYZT and YZfd ; of these four parts the 



ROGERS' DRAWING AND DESIGN. 



173 



M 



^ 




A 




A 




-^y 




^\ 




y\_ 




sy/^ 




/\U^<J= 





hTI 




Fig. 288. 



two larger parts, AKSI and YZfd are equal. The 
same is true of the two remaining smaller parts, 
KXTS and XYZT. 

To lay out these parts in the elevation a right 
angle abc is drawn, the sides of which intersect at 
right angles, the two largest branches of the joint. 
It is evident that the point b must be equidistant 
from both pipes. 

The right angle abc is divided first into three 
equal parts and then each one of these parts is 
divided in turn into two equal parts ; the right 
angle is thus divided into six equal parts, of which 
Kba is one part, KbX equals two parts, XbY equals 
two parts and Ybc one part. It will be noticed 
that this construction does not depend on the 
diameter of the pipe. 

The problem of developing the four-part elbow 
resolves itself into developing two only of its parts, 
one large branch and one smaller part of the elbow, 
the remaining parts being correspondingly equal to 
these. 

The circumference of the pipe. Fig. 288, is di- 
vided into sixteen equal parts by the points i, 2, 
3. 4, 5. etc. 

Through these points are drawn lines parallel to 
the center line of the pipe which is to be developed. 



174 



ROGERS' DRAWING AND DESIGN. 




g h ' is' g- / 

Fig. 289. 



C 



lu- 

In Fig. 289 the vertical branch of 

the elbow, AKSI, will be taken up 
for the purpose. The parallels upon 
the surface of this branch are AK, 
BL, CM, DN, EO, FP, GQ, HR 
and IS. Through the points K, L, 
M, N, O, P, Q, R and S draw par- 
allels for the part KXTS, which will 
be next developed ; some of these 
parallels are ST, RU, OV, PW. 

To develop the vertical branch of 
the four-part elbow set off, upon a 
straight line aa', Fig. 289, sixteen 
equal parts, which altogether are 
equal to the circumference of the 
*' cylinder, which is to be developed. 
Let the division points, a, b, c, d, 
e, f, etc., correspond to the division 
points, I, 2, 3, 4, etc., upon the circle, 
Fig. 288. Through the points, a, c, 
b, d, e, etc., draw vertical lines equal 
to the parallel lines drawn upon the 
surface of the vertical branch of the 
joint ; thus aj is made equal to AK 
(Fig. 288) bk equal to BL ; cl equal 
to CM and so on until ri is made 
equal to SI (Fig. 288). 



S 

J' 



ROGERS' DRAWING AND DESIGN. 



175 




The part laid out so far is ajtclmnopgri. This is one-half of the develop- 
ment ; the other half, irj'a' being exactly the same as the first one, may be 
laid out in the same way. 

The part tt'ss' is the development of the small part of the elbow. It is 
evident that its length, ts, must be equal to the circumference of the pipe 
in the elbow. The lines in the pattern, tt'ss' drawn at right angles to the 
center line of it, and bisected by it, are made equal to the parallel lines, ST, 
RU, OV, PW, etc., drawn 
upon the surface of the part, 
KXTS, Fig. 288. 

It is plain that the part, 
uu'vv' is equal to the part 
tt'ss', with the difference that 
the small parallels in it are laid 
out above the large parallels 
in the other part ; in the same 
manner, the pfe.rt yy'ww' is 
equal to the part aja'j'. 

Laying out the pattern in 
this manner makes it possible 
to cut out the complete elbow 
from one square piece of met- 
al, aya'y'. The spaces between 
the patterns are left for laps, 
which are necessary for join- 
ing all parts. 



176 



ROGERS' DRAWING AND DESIGN. 



To develop the dome of a boiler. 

A part of the boiler is shown by A, Fig. 290, to 
this the dome, B, is fastened by riveting. The 
illustration shows also the side view of the boiler 
and dome ; the top view of the dome C is drawn 
above the dome in the side view. 

This problem is exactly the same as the one ex- 
plained on page 170, where the intersection of two 
cylinders, of different diameters, was considered ; 



respectively to the parallel lines drawn upon the 
surface of the dome, Fig. 290. 

The dotted lines around the development show 
the lap which must be allowed for fastening the 
dome to the boiler by riveting. 

To develop the slope sheet of a boiler, shown in 
Figs. 2g2 and 2gj, 

The slope sheet which is to be developed is shown 
in Fig. 293 by ABCD. This sheet is of an irregular 



Fig. 291. 



the method emploved for the development in this 
case is, consequently, the same as in the previous 
problem. 

The circle C is divided into any number of equal 
parts, through the divisions of v/hich parallel lines 
are drawn on the surface of the dome. 

The development is shown in Fig. 291 ; its length 
is equal to the circumference of the dome. The 
parallel lines in the development are made equal 



shape. The side view is shown drawn to a larger 
scale by EF4'G in Fig. 294; the same figure shows 
one-half of the end view of the same slope sheet 
UW4. 

To prepare for the development divide the arc 
U4 into any number of equal parts, say four; 
horizontal lines drawn through the divisions, i, 2, 3 
will cut the line F4' at the points i', 2', 3' ; through 
these points draw lines parallel to G4' — these lines 



ROGERS' DRAWING AND DESIGN. 



177 



are 3'M, 2'N and I'O — -draw the line FP, parallel to 
these lines. Through point L on this line draw the 
line LH perpendicular to PF and perpendicular to 
all the slanting parallel lines just drawn ; these lines 
are cut by the perpendicular at the points H, I, J, 
K and L. 

From the point I on the line M3' lay off the 
distance 10 equal to 3Z ; from the point J on the 



through 



Draw a horizontal line aa', Fig. 295 
the point j near the center of this line draw kl 
perpendicular to it. On each side of j, on the line 
aa' lay off the distance ij and ji' each equal to HQ, 
Fig. 294. Then lay off the .distances if and i'f on 
the same line, each equal to QR, Fig. 294. 

Next lay off fe and f'e', each equal to RS, Fig. 
294, and lay off the distances eb and e'b' each equal 





FlO. 21)2. 



Fig. 293. 



line N2' lay off the distance JR equal to 2X. From 
the point K. on the line O i ' lay off the distance KS 
equal to Vi and from the point L on the line PF 
lay off the distance LT equal to UW ; through the 
points thus obtained draw the curve HORST. It 
is now possible to draw the development of the 
slope sheet. 



to ST, Fig. 294 ; through the points, b, e, f, i, j, i', 
f, e', and b' draw lines perpendicular to the line aa'. 
Lay off the distances jk and jl, equal to GH and 
H4' respectively, Fig. 294 ; then lay off the distances 
ih equal to h'i' and iq equal to i'q'— the distances hi 
and iq being equal respectively to the distances MI 
and 1 3', in Fig. 294. 



178 



ROGERS' DRAWING AND DESIGN. 



In the same manner the distances NJ and J 2', 
Fig. 294, are laid off on the lines gfpand g'f p' ; the 
distances OK and Ki', Fig. 294, are laid off on the 
lines deo and d'e'o' and the distances PL and LP", 
Fig. 294, are laid off in the same way, on the lines 
cbn and c'b'n'. 



and m' by two other arcs, drawn from the points c 
and c' as centers, with a radius equal to the dis- 
tance EP, Fig. 294. 

Join the points m and c, m and n, do the same at 
c'm' and n'm' and in this manner the pattern or 
templet of the slope sheet is obtained. 




Fig. 294. 



Now trace a curve through the points c, d, g, h, 
k, h', g', d' and c' and another curve through the 
points, n, o, p, q, 1, q', p' o' and n'. 

From the points n and n' as centers, describe arcs 
with a radius nm equal to n'm' each equal to EF, in 
Fig. 294 and intersect these arcs in the points m 



The dotted lines around the templet show the lap 
which must be allowed for riveting. 



ROGERS' DRAWING AND DESIGN. 



179 




Fig. 295. 



WORKING DRAWINGS. 



The purpose of a working drawing is to give the shopman information necessary to be known 
in order to construct the object or mechanism which is represented in the drawing. 

The drawings of the different parts of the machine are called " detailed drawings ;" in these 
each detail is represented in the most unmistakable manner, with all the dimensions of the parts written 
in, containing also, all further information concerning the part In question, that may be important for 
the purpose of making the patterns or forglngs. 

The drawings of the complete machine are called general drawings, or general plans, or 
"assembled drawings;" they show the whole arrangement of the machine, indicating the relative 
position of its parts ; they may also be made to show the motions of the movable parts. 

In preparing a detail drawing the first point is to decide the number of views required to Illus- 
trate the shape of the object and its parts in a complete and at the same time in a simple and 
easily understood manner. 

After deciding on the manner of views, the selection is decided upon of such a scale that will 
enable the placing of all the required views of the object within the space of the paper. 

As to the number of views required for an object no definite rule can be laid down ; it Is 
dependent upon the form and character of the figure and must be decided by the best judgment of the 
draughtsman. 

After ascertaining the most important dimensions of the mechanism a general drawing of the 
whole should be executed, omitting the smaller parts ; after this particular drawings are made. The 
larger and more important parts are first produced, next, the smaller parts which are to be attached to 
the larger parts. 

The materials of which the parts are to be made should be shown either by sections or by 
special remarks, notes, etc.; the methods of work which are to be followed by the workman should be 
indicated, sometimes to the extent of pointing out what machine tool is to be used for the work. 

— 



184 



ROGERS' DRAWING AND DESIGN. 



DIMENSIONING DRAWINGS. 

This interesting subject has been referred to and 
illustrated upon pages 132-134; in these the stu- 
dent will find much valuable matter relating to the 
subject. 

Putting the dimensions on a drawing correctly is 
not only one of the most important but also most 
difficult parts of the work of a draughtsman ; the 
latter will put in those dimensions only which will be 
required by the shopman; the manner in which this 
is done must depend upon the method to be used 
by the workman in constructing the part to which 
the dimensions refer; for this reason, an acquaint- 
ance with the methods adopted in shop practice as 
well as with the tools to be used is essential. 

Every dimension necessary to the execution of 
the work should be clearly stated by figures on the 
drawing, so that no measurements need to be taken 
in the shop by scale. All measurements should be 
given with reference to the base or starting point 

Note. — It must be understood that the scale on a drawing is not 
given for a shopman to take his dimensions from ; such dimensions 
must all be taken from the dimension figures ; the scale is given for 
the chief draughtsman's use, or whoever may check the drawing, and 
also for the use of other draughtsmen who may make at some future 
time alterations or additions to the drawing. 



from which the work is laid out, and also with 
reference to center lines. 

All figured dimensions on drawings must be in 
plain, round vertical figures, not less than one-eighth 
inch high, and formed by a line of uniform width 
and suiificiently heavy to insure printing well, omit- 
ting all thin, sloping or doubtful figures. All 
figured dimensions below two feet are best ex- 
pressed in inches. 

It may be put down as a rule that the draughts- 
man must anticipate the measurements whi'ch will 
be looked for by the workman in doing the w(^fk, and 
these dimensions only must be put on the drawing. 

Surfaces which are to be finished should be 
plainly marked " finished." When a particular tool 
or machine which is to be employed in finishing is 
mentioned, by putting the name of the tool, in small 
letters, near the surface which is to be finished, the 
word " finished " need not be added. 

When an object is to be turned in a lathe, the 
dimensions of the turned surfaces must be given by 
their diameters. Near the outline of the surface 
the word " turned " should be plainly marked. For 
some purposes it may be desirable to put in the 
radius for turned work ; if such a case may be fore- 
seen by the draughtsman, the dimension should be 
inserted. 



ROGERS' DRAWING AND DESIGN. 



185 



Wherever the ends of a piece of work in the 
lathe are to be finished, as, for instance, the two 
ends of a hub on a pulley, the word "face" should 
be plainly marked near the surface which is to be 
finished in this manner. 

The dimensions zuritten on the drawifig should 
always give the actual finished sizes of the object, no 
mattej' to what scale the object may be drawn. 

All dimensions which a shopman may require 
should be put on a drawing, so that no calculation 
be required on his part. 

For instance, it is not enough to give the lengths 
of the different parts of the object, but the length 
over all, which is the sum of all these lengths, 
should also be marked as shown in Fig. 296. 

The figures giving the dimensions should be placed 
on the dimension lines, and not on the outline of 
the object. 

The dimension lines should have arrow heads at 
each end and the points of these arrow heads should 
always touch exactly the lines, the distance between 
which is indicated by the dimension as illustrated in 
Fig. 297. 

The figure should be placed in the middle of the 
dimension line at right angles to that line, and so 
as to read either from the bottom, or from the right 



hand side of the drawing. The arrow heads should 
be put inside of the lines, from which the distance, 
as given in the dimension, is reckoned. 



•% 



VZ2^ZZ^ 




When the space between these lines is too small 
for the figures, the lines being very close together, 
as shown in Fig. 298, the arrow heads may be 



186 



ROGERS' DRAWING AND DESIGN. 



placed outside and the figures also be put outside, 
in which case an arrow should be put in to indicate 
the proper position of the figures. 

The dimension lines should also be put in the 
drawing, very near to the spaces or lines, to which 
they refer. 

When "the view" is complicated, dimension lines 
drawn within it, might tend to make it still more 
obscure and difficult to understand ; in such a case 



4" 



^ 'V 



_L" 

4 



Fig. 297. 



Fig. 298. 



the dimension lines should be carried outside of the 
view and extension lines drawn from the arrow 
heads to the points, between which the dimension is 
given. 

When the dimension includes a fraction, the nu- 
merator should be separated from the denominator by 
a horizontal line and not by an inclined line ; care 
should also be taken to write the figures in a very 
clear and legible manner and crowding should be 
avoided. 



When a dimension is given in one view, it need not 
be repeated in another view, except when such a rep- 
etition is essential to locate the size in question. 

For must shop drawings blueprints are used ; the 
beginner will find that the dimensions on the print 
do not cc rrespond with the scale ; this is due to the 
shrinkin'3^ of the blueprint paper after it has been 
washed ; as the dimensions on a blueprint are gen- 
erally shorter than the scale by which they have 
been meiisured. '^ 

In many shops there exists a rule ; that every 
draughtsman must mark plainly on his drawing in 
some place where it is easily seen by the workman, 
" do not scale drawing." 

When a drill is to be used, it is advisable to write 
near the hole in question the word ' drill." Should 
the hole be provided with a thread to be produced 
by a. tap, write " tap," adding to this one word, its 
size or number. 

When the hole is of a comparatively large diam- 
eter, so that it can be finished only in the lathe by 
turning, or in the boring mill by boring, the words 
"turn" or "bore" respectively should be put in 
near the circumference of the hole. In this case, 
also, the diameter of the hole, and not the radius, 
should be given. 



ROGERS' DRAWING AND DESIGN. 



187 



When a number of holes are to be laid out in 
one piece of work, the distance from center to 
center should be given, and not the distances be- 
tween the circumferences of the holes. 

When a number of holes are at equal distances 
from a central point, or when their centers are lo- 
cated in the circumference of a circle, this circum- 
ference should be drawn through the centers of the 
holes, and the diameter should be given as a dimen- 
sion. The distances between the centers of the 
holes measured on a straight line, or measured as a 
part of the circumference, on which their centers 
are located, should also be noted. 

In practice, at times, instead of dimensions refer- 
ence letters are used, thus : 







e -L- ^ 






)K 


^T? 




^m 




k 


1 

1 ^ ^ 


5-T- 


------t 


i 


a 




^ 


^J 




Fig. 299. 



D— =diam. of shaft, 2j4 inches. 
L=length of bearing, 3-)^ inches. 
T=thickness of collar, J'^g inch, 
d^diam. of collar, 3^ inches. 



It is preferable to give the diameters of turned 
and bored work on a section, instead of an end 
drawn separately ; confusion is sometimes caused 
by a number of radial dimensions. 

The following are quoted from A. W. Robinson's 
admirable Office Rules : 

" Every drawing, whether whole or half-sheet, 
shall have the title, date, scale and number of the 
sheet stamped in lower right-hand corner, and the 
quarter and eighth sheets printed on top. 

" The name of the drawing, as given in the title, 
is invariably to consist of two divisions in one line 
separated by a hyphen. The first division is to 
state the general name of the thing or machine, and 
the second name is to clearly designate the part 
or parts represented (or if a general view should so 
state). The wording of titles should be submitted 
to the chief engineer or head draughtsman for ap- 
proval. 

" Each drawing shall bear the name of the 
drauofhtsman and examiner, the surname being- used 
without initials. 

" Drawings of piping details should be made in 
diagram form, using standard symbols. 

"All detail parts for standard or repetition work 
shall be shown unassembled as far as possible." 



188 



ROGERS' DRAWING AND DESIGN. 



TINTS AND COLORS. 

It is sometimes found necessary to prepare a 
highly finished and shaded drawing of the work in 
hand, and for special purposes, they are also tinted 
and colored ; such elaborations, in fact, are much 
admired by the uninitiated, although no criterion as 
to the scientific value of the object is represented. 

Mechanical drawings are seldom tinted, but are 
mainly produced in India ink. Where, however, a 
fine effect is desired, working drawings are colored, 
so as to show at a gflance the material of which the 
different parts are to be made. 

The colors required are few but should be of the 
best quality. Besides India ink the following water- 
colors are generally used : 

I, Neutral-tint; 2, Prussian Blue; 3, Chrome 
Yellow ; 4, Gamboge ; 5, Raw Sienna ; 6, Carmine ; 
7, Vermilion; 8, Vermilion Red; g. Sepia; 10, 
Indigo. These come in hard cakes. 

Certain colors and tints represent different metals 
and materials as follows : 

Wrought Iron — Pruss'an Blue. 

Steel — Carmine and Prussian Blue, mixed to give 
a purple shade. 

Steel Casting — Same as the above darkened by 
Venetian Red. 



Cast Iron — Neutral-tint made of India Ink, In- 
digo, mixed with a little Carmine. 

Brass — Gamboge or Chrome Yellow. 

Babbitt — Emerald Green • sometimes light mix- 
ture of India Ink. 

Copper — Purple Lake. 

In applying the tints, the paper must be ex- 
panded and stretched evenly all over its surface ; 
otherwise when the moist tint is applied the paper 
will wrinkle and get out of shape ; to do this cut 
the paper at least half an inch less in size than the 
drawing board ; lay the paper face down, turn up a 
margin or edge of about three-fourths of an inch 
all round, then dampen the paper with a sponge 
and clean water ; allow it to soak for a few minutes 
until it is evenly and thoroughly dampened ; next 
turn the paper upside down (face up). 

Apply strong paste to the under side of the mar- 
gin all round ; rub down, on the drawing-board, 
working from the center of the board outwards so 
as to exclude the air and prevent creases or furrows. 
The board is then inclined and left to dry slowly ; 
make sure that the paper is all well pasted and 
every part of the edges attached to the board. 

If tracings are required to be tinted or shaded, 
the color may be applied before the tracing is cut 
off, or what is more usual, the color may be applied 
on the back of the tracing ; then there is no liability 
to wash out the lines. 



ROGERS" DRAWING AND DESIGN. 



189 



TRACING AND BLUE PRINTING. 

Whenever it is desired to have more than one 
copy of a drawing, a "tracing" is made of it and 
from this as many blueprints can be obtained as 
are required. 

When a tracing is needed for making blueprints 
a piece of tracing paper or tracing cloth of the 
same size as the drawing is placed over the original 
drawino- and fastened to the board. This tracing 
paper or cloth is almost transparent ; the tracing is 
a mechanical copy of a drawing made by repro- 
ducing its lines as seen through a transparent 
mf-dium such as has been described and the lines of 
the drawing can be seen through it. 

The surfaces of the tracing cloth are called the 
"glazed side" and the "dull side," or "front" and 
" back ;" the glazed side has a smooth polished sur- 
face and the dull side is like a piece of ordinary 
linen cloth. 

Note. — Many concerns have rules of their own, directing their 
ilraughlsnien to use either the smooth or the rough side for all pur- 
poses ; if there are no such rules, it is left to the judgment of the 
draughtsman. 

While it is immaterial which side of the cloth is used in tracing, 
however, if any mistakes are made and have to be corrected this can be 
done easier on the glazed side ; on the contrary, if any additiuns must 
be made to the tracing, which have to be drawn in pencil first, the dull 
side will be found most convenient, as the pencil marks show plainer 
on the dull side. 



Drawing on tracing paper or cloth is effected by 
pencil and drawing pen as in ordinary work. 

The tracing cloth must be fastened to the board, 
over the drawing, by pins or thumb tacks ; moisture 
or dampness should be carefully avoided and the 
drawing done, preferably, on the smooth side of the 
cloth. 

When tracing cloth will not take ink readily a 
small quantity of pounce may be applied to the sur- 
face of the cloth and distributed evenly with a piece 
of cotton waste, chamois, or similar material, but 
the pounce should be thoroughly removed before 
applying the ink. 

In making tracings the order to be followed is as 
follows: I, ink in the small circles and curves; 2, 
ink in the larger circles and curves ; 3, then all the 
horizontal lines, beginning at the top of the draw- 
ing and working downward ; 4, next ink in all the 
vertical lines, commencing at the left and moving 
back to tht' right ; 5, draw in the oblique lines ; 6, 
in finishing the figuring and lettering should be 
done with India ink, thoroughly black. 

" Erasing," in case of mistakes or errors, should 
be done with an ink-eraser or a sharp, round erasing 
knife ; the surface of the tracing cloth must be 
made smooth in those places where lines have been 
erased ; this is accomplished by rubbing the cloth 



190 



ROGERS' DRAWING AND DESIGN. 



with soapstone or powdered pumice stone, applied 
with a soft cloth or with the finger. When a mis- 
take made is so serious that it cannot be corrected 
by erasing, a piece of the tracing cloth may be cut 
out and a new one inserted in its place. 

A finished tracing should be provided Avith the 
title of the drawing, the date, scale and the initials 




Read this way. 

Fig. 200. 

of the draughtsman as shown in Fig. 302 which is a 
representation of a blueprint taken from a tracing. 
The letters on a drawing should be placed, as near 
as possible, as shown in Fig. 300 ; when a part to 
which a note refers is at an angle to the base line, 
the letters should be placed parallel to the part 
mentioned. All the letters should be so placed that 
the drawing can be " read " without turning the 
drawing completely around. 



BLUE PRINTS. 

As many copies as may be desired of a tracing 
can be made from it by the process known as " blue 
printing." 

In order to make serviceable blue prints three 
things are essential: i, good paper; 2, proper 




Fig. 301. 

chemicals for coating the paper, and 3, a good print- 
ing frame. One form of the latter is shown in Fig. 
301, with half of the back raised ; the back is made 
in two sections and hinged together, this being 
done in order to enable the operator to lift one-half 
of the back and inspect the prepared paper, so as to 
ascertain if the print is of the right color. 



ROGERS' DRAWING AND DESIGN. 



191 



The springs shown in the figure are intended to 
keep the hinged back pressed close against the pre- 
pared paper, tracing and £-/ass, — the latter, of course 
being invisible in the cut, but which should consist 
of good clear, double thick glass of a size to fit the 
frame. 

The inside surface of the back or that side which 
presses against the prepared paper is always covered 
by felt or three or four layers of Canton cotton, 
which are glued to it. 

The "printing" is very simple, as the only appa- 
ratus necessary is a blue printing frame one form of 
which is shown in Fig. 301, and a trough containing 
water for washing the prints ; in brief, this is the 
method of procedure : i, the tracing is fixed in the 
frame, with the surface on w-hich the drawing is 
made, i. e., the inked side, next to the glass ; 2, the 
"sensitized" side of the paper is next laid against 
the back of the tracing ; 3, replace the wooden 
cover and fasten down the springs so that both 
paper and tracing are compressed firmly against 
the glass, permitting no creases or air spaces be- 

NoTE. — Large printing frames are generally mounted on a frame, 
which are provided with wheels running on rails ; by means of this ar- 
rangement the frame can be pushed out through a window for exposure. 



tween them. This should be done in a darkened 
room ; 4, expose for three to six minutes, according 
to the intensity of the sun ; 5, the sensitized paper 
is to be taken out of the frame and quickly washed 
in clean cool water ; the drawing will appear in white 
lines on blue ground ; 6, the print finally hung up 
by one edge so that the water will run off and the 
print allowed to dry. 

The sun rays or a strong electric light act upon 
those parts of the sensitized paper not covered by 
ink lines in the drawing and change their yellowish 
color gradually to a dull gray. When taken out of 
the frame and washed, as explained already, the 
paper turns dark blue leaving the lines of the draw- 
ing in plain white. 



TEST PIECES. 

To make good blueprints, being guided only by 
the appearance of the exposed edge of sensitized 
paper, requires considerable experience. Very often, 
especially on a cloudy day, the edge looks just 
about right, but when taken out of the frame and 
given a rinsing, it is only to find that the print looks 



192 



ROGERS' DRAWING AND DESIGN. 



pale because it should have been allowed to remain 
exposed for a longer period. 

Now simply take a small test-piece of the same 
paper (say about 4 inches square) and a piece of 
tracingf cloth with several lines on its surface and 
lay these small pieces out at the same time the real 
print is being exposed, and cover these samples 
with a piece of glass about 4 inches square. As a 
general rule, we can find a place on top of the frame 
for the testing-piece, and by having a small dish of 
water at hand for testing the print by tearing off a 
small bit and washing same to note its appearance, 
the novice can get just as good results as the exper- 
ienced hand and without much danger of failure. 



HELIOGRAPHIC PRINTING. 

To obtain sharp lines on a blueprint all lines on 
the tracing should be made heavier than on ordi- 
nary drawing paper and a sharp- inking pen should 
be used. 

Paper which has a glossy or starched appearance 
should never be used, as the blueprint solution when 
applied to the paper will intermix with the starch 
and the result will be poor prints. Drawing paper 
or blueprint paper (unprepared) which may be 
obtained of any dealer will give the best results. 



The sensitized paper is sold ready for use, but it 
can be prepared by dissolving two ounces of cit 
rate of iron and ammonia in eight ounces of soft 
water; keep in a dark bottle; also, one and one- 
third ounces of red prussiate of potash in eight 
ounces of water ; keep in another dark bottle ; when 
about to use mix an equal quantity of each in a cup 
and apply in a dark room with a soft brush or 
sponge to one side of white rag paper, similar to 
envelop paper. To complete the process let it dry 
and put away in a dark place until required for use. 

When several prints are to be made the second 
one may be placed into the frame while the first 
one is soaking ; when the print is properly soaked, 
say about ten minutes, lift it slowly out of the water 
by grasping two of its opposite corners ; immerse 
again and pull out as before. This is to be con- 
tinued until the paper does not change to a deeper 
blue color. Hang the paper on the rack by two of its 
corners to dry. In case any spots appear it is an 
indication that the prints were not properly washed. 

When corrections or additions are to be made to 
a blueprint a special chemical preparation must be 
used to make white lines. A solution of quick- 
lime and water is generally used for this purpose. 
When white lines or figures are to be obliterated a 
blue pencil may be used to cover same. 



ROGERS' DRAWING AND DESIGN. 



193 



BLACK PROCESS COPYING. 

This is accomplished by specially sensitized paper 
by which a fac-simile of the original drawing can be 
made ; that is, black lines upon white ground ; this 
method of printing avoids the objection to the blue- 
print paptr of shaded drawings which show light and 
shade reversed. 

The prints made by this process are said to be 
permanent and can be altered, added to or colored 
the same as original drawings. 



SENSITIZING. 

This term is much used in photography and means 
to make "sensitive" to the action of light derived 
from the sun or from electricity. To sensitize blue 
printing paper proceed as follows : The paper 
should be white, smooth and of good quality, it is 
best to purchase such paper as is purposely made 
for sensitizing. 

The solution used for ordinary blue printing is 
made according to the following receipt : 

a. One ounce of red prussiate of potash dissolved 
in 5 ounces of water. 

b. One ounce of citrate of iron and ammonia dis- 
solved in 5 ounces of water. 



Keep the solutions separate in dark colored bottles 
in a dark place not exposed to the light. To pre- 
pare the paper, mix equal portions of the two solu- 
tions and be careful that the mixtures are not long-er 
exposed to the light, than is necessary to see by. It 
is, therefore, a necessity to perform this work in a 
dark room, provided with a trough of some kind to 
hold water ; this should be larger than the blue- 
print and from six to eight inches deep ; a flat board 
should be provided to cover this trough ; there 
should also be an arrangement like a towel rack to 
hang the prints on while drying. The sheets should 
be cut in such a manner as to be a little larger than 
the tracing, in order to leave an edge around it 
when the tracing is placed upon it. From ten to 
twelve sheets are placed upon a flat board; care 
must be taken to spread them flat one above an- 
other, so that the edges are all even. The sheets 
should be secured to the board by a small nail 
through the two upper corners, strong enough to 
hold the weight of the sheets when the board is placed 
vertically. 

Place the board on the edges of the trough with 
one edge against the wall and the board somewhat 
inclined, only as much light as is absolutely required 
should be obtained from a lamp or gas jet, turned 
down very low. The solution referred to above 



194 



ROGERS' DRAWING AND DESIGN. 



should be applied evenly with a wide brush or a fine 
sponge over the top sheet of paper. When the top 
sheet is finished remove it from the board by pulling 
at the bottom of same and tearing it from the nail 
which holds it ; place said sheet in a drawer where 
it can lie flat and where it cannot be reached by the 
light. 

Treat the remaining sheets in the same way as 
the first one. 

In place of blue printing paper bj'ozun printing 
paper may be substituted. After an exposure of 
five minutes in bright sunlight, the margin protrud- 
ing beyond the tracing cloth changes its original 
light yellow color to a dull reddish brown. The print 
is then immersed in the water-bath and thoroughly 
soaked and rinsed on both sides ; the back ground 
immediately changes to a brown color the lines com- 
ing out in perfect white. The prints are then placed 
in a fixing solution and washed again during fifteen 
or twenty minutes. 



Note. — There is a method of copying drawings on thick paper and 
even on cardboard ; it consists of using a kind of sensitive paper known 
as " gelatine " or "bromide"; this is covered with a sensitizing com- 
pound made chiefly from the bromide of silver put on in a layer of 
gelatine. 



TO SENSITIZE PAPER FOR BLUE LINES 
ON A WHITE GROUND. 

The following process, credited to Captain Abney, 
yields a photographic paper giving blue lines on a 
white ground : 

Common salt 3 ounces. 

Ferric chloride 8 ounces., 

Tartaric acid 35^ ounces.-' 

Acacia 25 ounces. - 

Water 100 ounces. 

Dissolve the acacia in half the water, and dissolve 
the other ingredients in the other half ; then mix. 

The liquid is applied with a brush to strongly- 
sized and well-rolled paper in a subdued light. The 
coating should be as even as possible. The paper 
should be dried rapidly to prevent the solution sink- 
ing into its pores. When dry, the paper is ready 
for exposure. 

In sunlight, one or two minutes is generally suf- 
ficient to give an image, while in a dull light, an hour 
is necessary. 

To develop the print, it is floated immediately 
after leaving the printing frame upon a very weak 



ROGERS' DRAWING AND DESIGN. 



195 



solution of potassium ferrocya^iide. None of the 
developing solution should be allowed to reach the 
back. The development is usually complete in less 
than a minute. The paper may be lifted off the 
solution when the face is wetted, the development 
proceeding with that which adheres to the print. t\ 
blue coloration of the backofround shows insufficient 
exposure, and pale-blue over-exposure. 

When the development is complete, the print is 
floated on clean water, and after two or three minutes 
is placed in a bath, made as follows : 

Sulphuric acid 3 ounces. 

Hydrochloric acid 8 ounces. 

Water loo ounces. 

In about ten minutes the acid will have removed 
all iron salts not turned into the blue compound. It 
is next thoroughly washed and dried. Blue spots 
may be removed by a 4 per cent, solution of caustic 
potash. 

The back of the tracing must be placed in contact 
with the sensitive surface. 

Note. — The sensitized paper, when not in use, should be kept in a 
dark, dry and air-tight place, as with age and exposure the paper 
becomes deficient in quality ; the best way to preserve the sensitized 
paper is to have made a tin c^dinder about 3>2 inches in diameter and 
an inch or two longer than the paper it is desired to keep and with a 
tight cover to fit over the outside at one end. 



MOUNTING BLUE PRINTS FOR THE 

SHOP. 

The shop foreman is often put to a great deal of 
inconvenience because of the rapid destruction, 
either through becoming soiled or torn, of blueprints 
which are used at the machines. Some damage is 
undoubtedly due to careless handling of the prints, 
but the greater part of the wear and tear cannot be 
avoided, even with the greatest care, and the spot- 
ting and creasing soon make the print unusable. 

To obviate this the blueprints can be fastened on 
common sheets of pasteboard, but in time the paste- 
board itself becomes broken and oil-spotted, hence 
the frequent adoption of the idea of using thin sheet 
iron as a backing. 

The prints in common use in the shop are first 
pasted on pieces of sheet iron, then both sides are 
varnished over, so as to make the paper oil and 
waterproof. After being subjected to this treatment, 
the prints can be hung up near the machines. By 
thus mounting the prints they are clean and clear, 
and can be filed away in a small space when not in 
use ; moreover, they are practically indestructible, 
because when soiled they can be put under the hose 
and washed off. 



196 



ROGERS' DRAWING AND DESIGN. 



Sheets that are likely to be removed and replaced, 
for any purpose, as working drawings generally are, 
can be fastened very well by small copper tacks, or 
the ordinary thumb-tacks, driven in along the edges 
at intervals of 2 inches or less. 

The paper can be very slightly dampened before 
fastening in this manner, and if the operation is 
carefully performed the paper will be quite as 
smooth and convenient to work upon as though it 
were pasted down ; the tacks can be driven down so 
as to be flush with, or below the surface of, the 
paper, and will offer no obstruction to squares. If 
a drawing is to be elaborate, or to remain long 
upon a board, the paper should be pasted down. 

To do this, first prepare thick mucilage, and have 
it ready at hand, with some slips of absorbent paper 
I in. or so, wide. Dampen the sheet on both sides 
with a sponge, and then apply the mucilage along 
the edge, for a width of ^ -s/g in. It is a matter of 
some difficulty to place a sheet upon a board ; but 
if the board is set on its edge, the paper can be 
applied without assistance. Then, by putting the 
strips of paper along the edge, and rubbing over 
them with some smooth hard instrument, the edc/es 

o 

of the sheet can be pasted firmly to the board, the 
paper slips taking up a part of the moisture from 
the edges, which are longest in drying. 



TO MAKE DRAWINGS FROM THE PRINTS. 

To accomplish this, the blueprints may be inked 
over with " waterproof ink " and when thoroughly 
dry washed with a solution of oxalate of potash, 
treated thus the ink lines will remain, and the blue 
ground will fade and become white and appear 
similar to an original drawing ; the prints can be 
bleached by washing them in a saturated solution of 
oxalate of potash, as above. 



MECHANICAL DRAWING AND ITS RELA- 
TION TO PRACTICAL SHOP WORK.* 

The relation of the drafting room to practical 
shop work is a vital subject that is constantly forced 
upon the attention of all by the occurrences of daily 
work, but each department, the drafting room and 
the shop, has its well-defined place. 

In mechanical work we must have first the zdea, 
or conception of what is wanted, whether the idea 
comes from the inventor, the draughtsman or the 
machinist;the draughtsman, by means of the drawing, 
becomes the interpreter of the idea to the shop. 

*NoTE. — From an address delivered by L. D. Burlingame, Chief 
Draugtitsman at ttie Brown & Sliarpe Manufacturing Company, before 
the Eastern Manual Training Association, at the coiivention held at 
Boston. 



ROGERS' DRAWING AND DESIGN. 



197 



The three important relations hereafter dwelt 
upon can be stated briefly thus : 

First — The drafting department as the interpreter 
/^ the shop —the drawing making plain the meaning 
and requirements of the designer to the workman. 

Second— The drafting department as the inter- 
preter of the shop — the draughtsman, through con- 
sultation and discussion, making available the 
practical experience and suggestions of the shop 
man. 

Third — The drafting department as the recorder 
for the shop- — the records of all data and information 
being so compiled and kept as to be reliable, and 
quickly available when needed. 

First. Let us consider the drafting room as an 
interpreter to the shop : 

In preparing drawings each piece must be fully and 
separately detailed, and in many shops, each on a 
sheet by itself; all particulars of oiling and venting 
oil holes must be shown, grinding limits given, the 
depths of tapped holes figured. There must be an 
indication of when stock is to be allowed for fitting, 
and of the special kinds of finish on machined sur- 
faces. All special tools used in manufacturing the 
piece must be listed below its name, and perhaps 
a list of operations given either on the drawing or 
in a separate list. 



Second. I would earnestly recommend that there 
be instilled into the minds of technical students the 
importance of taking advantage of the great mass 
of mechanical knowledge and the ideas stored up in 
the minds of the mechanics of the country, in the 
minds of the men that are actually doing the work, 
and that the students have it impressed upon them 
that if they become draughtsmen, one of their im- 
portant duties will be so to get in touch with the 
shop as to make this knowledge available, even 
though it may come to them in crude form from 
a mind not trained to analyze, to classify and to put 
ideas upon paper — in other words, that they learn 
to be the interpreters of the shop. 

Third. Briefly the ofhce of the drafting room is 
the recorder for the shop ; here we touch upon the 
important work of tabulating, listing and classify- 
ing ; for example : thousands of special tools accumu- 
late in a large shop ; pro ninent among these are 
taps, reamers, drills and counterbores, cutters, gauges, 
etc. ; there are many things to be preserved for ref 
erence that naturally find their way to the drafting 
room, such as trade catalogues, photographs, copies 
of patents and technical journals. The treatment of 
these in indexing makes all the difference whether 
they are valuable — of growing value as time goes 
on — or nearly as worthless. 



198 



ROGERS' DRAWING AND DESIGN. 



The importance of the shop side has perhaps been 
emphasized in what has been said, but I certainly 
would not belittle the draughtsman, even aside from 
the high position he often holds as designer and 
constructor. I agree with the statements made by 
Prof. Charles L. Griffin; he says: ''The workman 
of to-day is not permitted to assume dimensions or 
shapes ; it is his business to execute the draughts- 
man's orders ; it is, however, often his privilege to 
choose his own way of doing it, but further than this 
modern practice does not allow him to go. 
The drawing is supreme, it is ofificial ; it must be 
plain, direct and all sufficient." It might be added 
that to make it so the draughtsman must mentally 
put himself in the place of the shopman, and antici- 
pate his needs. The workman will then respect the 
draughtsman and his work, and will be willing to fol- 
low implicitly the instructions given on his drawings. 



TO READ WORKING DRAWINGS. 

A working drawing should be made, primarily, as 
plain as possible by the draughtsman ; second, the 
workman should patiently and carefully study it, so 
that it is thoroughly understood. 

In studying a drawing, the object it is intended to 
represent should be made as familiar as possible to 



the mind of the student, so that he may fill out in 
imagination the parts designedly left incomplete — as 
in a gear wheel where only two or three teeth are 
drawn in, that he may see, mentally, the whole. 

Drawings are almost always made "finished size," 
that is, the dimensions are for the work when it is 
completed. Consequently all the figures written on 
the different parts indicate the exact size of the work 
when finished, without any regard to the size of the 
drawing itself, which may be made to any reduced 
and convenient scale. 

Even in full size drawings this system of figuring 
is not objectionable. It is a system which should 
be followed whenever a drawing is made " to work 
to," for it allows the workman to comprehend at a 
glance the size of his work and the pieces he has to 
get made. Figuring makes a drawing comprehen- 
sible even to those who cannot make drawings. 

In some figures it is necessary to show end views, 
also section views, to enable all measurements to be 
read from the drawing. 

Fig. 302 represents a blueprint of a bracket-bear- 
ing, constructed from the drawing, for the Raabe 
compound oil engine ; the front, end view and plan, 
with dimensions, are shown ; the scale is full size 
1 2"== I ft. 



MACHINE DESIGN, 



The study of mechanical drawing not only consists in copying drawings of machinery and dia- 
grams by accurate measurements and fine finished lines, but it includes the purpose and practical 
operation of the mechanism designed ; i. e., drawing as a means to an end. 

The designing of machines requires an extended acquaintance of parts and of similar mechan- 
isms which have been found suitable for the work required and thus have become standard elements 
of construction ; to utilize this knowledge is ofte' the lite task of the draughtsman and designer of 
machinery. 

It is a matter of common acceptance that machine design depends more upon an acquaint- 
ance with mechanism and siiop practice than upon a knowledge of the strength of materials and other 
kindred subjects making up the science of mechanics; this is the reliance, however unscientific it 
may be, that is depended upon in perfecting the designs for the machinery that is being produced 
to-day, and there will probably never be another system that, on the whole, will be more satisfactory. 

It is, however, not sufficient to limit our education to observation of completed working 
machines ; it is just as necessary to know the theoretical principles and laws of mechanical con- 
struction ; these have been classified as Theoretical Mechanics or Theory of Mechanism ; a few 
necessary definitions and general considerations will be found on the succeeding pages 

Note. — "The correct forms to be given to the raatenais employed in the construction of tools or machinery depend entirely 
upon Liatural principles. Natural form consists in giving each part the exact proportion that will enable it to fuifiU -ts assigned duty 
with the smallest expenditure of material, and in placing each portion of the materials under the most favorable conditions of position 
that circumstances will admit of. 

" Such natural form is not only the most economical but, strange to say, it is always correct in every :'espect, and is invariably 
beautiful and lovely in its outlines." — Andrews. 

?05 



206 



ROGERS' DRAWING AND DESIGN. 



The most successful designer is no doubt born 
with a love for mechanics and a measure of inventive 
ability ; if to these inherent qualities be added a 
retentive memory, a mind trained to observe closely, 
deliberate carefully and decide wisely, he should be 
a success. Technical education in itself is of little 
avail ; but if allied to these other qualities, perfect 
and round them out, smoothing the way over places 
that would be otherwise well nigh insurmountable. 

The cost and results of special machinery depend 
so much on the ability of the designer that it may be 
well to consider what his attainments should be ; he 
should be able to clearly illustrate his ideas- — not nec- 
essarily a finished draughtsman — and he should have 
a practical experience in machine shop practice so that 
to know that the elements of his design can readily 
be machined, and that no unnecessary trouble be had 
with the patterns or in making castings from them. 

He should also know enough of machine design 
that no illy-proportioned parts disfigure or weaken 
the structure, and sufficient taste to realize that true 
art in machinery does not consist of imitating archi- 
tectural embellishments ; for beauty, as well as for 
strength and cheapness, castings should be of the 
simplest shape possible ; rounded corners, especially 
interiors, straight lines where permissible, with all 
projections provided for originally, rather than lo 



appear as afterthoughts, are the principal elements 
of mechanical beauty. 

In reference to the particular case in hand, the 
designer ought to familiarize himself with the 
methods before employed, if the product has been 
previously made ; quantities of product expected from 
the machine, space to be occupied, size, weight, 
speed, power required, and number likely to be made, 
should be carefully considered. 

All notes, deductions, sketches and the like should 
be carefully preserved, at least until the machine is 
completed ; that is, actually built, for these sketches 
may prove to be proof of the most convincing char- 
acter should questions arise as to mechanical elements 
considered, even at the time unapproved. 

It is inconceivable that without shop experience 
a designer can be highly successful ; the more ex- 
perience the better, not in one shop alone but 
several. To succeed requires determination and 
painstaking hard work ; a mistaken figure, a wrong 
calculation or blunder of any kind is sure to bring 
vexation for some one, and possibly a serious loss. 

Finally, always study simplicity of construction, 
avoiding as far as possible all special shaped 
wrenches, etc., and using "more than enough" of 
iron and steel in all designs, to assure strength and 
durability. 



ROGERS' DRAWING AND DESIGN. 



207 



DEFINITIONS AND GENERAL 
CONSIDERATIONS. 

Attraction. This is an invisible power in a 
body by which it draws anything to itself ; the power 
in nature acting naturally between bodies, or par- 
ticles, tending to draw them together ; the attraction 
of gravitation acts at all distances throughout the 
universe ; adhesive attraction unites bodies by their 
adjacent surfaces ; chemical attraction, or chemical 
affinity, is that peculiar force which causes elemen- 
tary atoms or molecules to unite. 

Co-ef&cient is a number expressing the amount 
of some change or effect under certain fixed con- 
dition as the co-efficient of expansion ; X\\^ co-efficient 
of friction ; the word generally means, " that which 
unites in action with something else to produce the 
same effect!' 

Cohesion is that force which binds two or more 
bodies together. It is that force which the neigh- 
boring particles of a body exert to keep each other 
together. 

Ductility \s that property by which some metals 
can be drawn out into wire or tubes. 

Mffort is a force which acts on a body in the 
direction of its motion. 



Elasticity is the property possessed by most solid 
bodies, of regaining their original form or shape, 
after the removal of a force which caused a change 
of form. 

^Energy is the capacity for performing work ; 
the kinetic energy of a body is the energy it has in 
virtue of being in motion ; kinetic energy is some- 
times called actual energy ; potential energy is energy 
stored up as that existing in a spring or a bent bow, 
or a body suspended at a given distance above the 
earth and acted upon by gravity. 

The efficiency of a machine is a fraction ex- 
pressing the ratio of the useful work to the whole 
work performed, which is equal to that expended. 

A Factor is one of the elements or quantities 
which when multiplied together form a product. 

Force is that which tends to produce or to de- 
stroy motion ; if a body is at rest anything which 
tends to put it in motion is a force ; centrifugal force 
is that by which all bodies moving around another 
body in a curve, tend to ffy off from the axis of their 
motion ; centripetal is that which draws, or impels a 
body toward some point as a center ; force is equiv- 
alent ^.o push or pull. 



208 



ROGERS' DRAWING AND DESIGN. 



Fatigue of Metals. In many cases materials 
are subject to impulsive loads and a gradual diminu- 
tion of strength is observed ; in part this deteriora- 
tion of strength may be due to the ordinary action 
of a live or repeated load, but it appears to be more 
often due directly to the gradual loss of the power 
of elongation in consequence of the slow accumula- 
tion of \\i^ permanent set ; the latter may be defined 
as the fatigue of metals. 

Friction is that force which acts between two 
bodies at their surface of contact so as to resist 
their sliding on each other, and which depends on 
the force with which they are pressed together. 

Gravity. We can not say what gravity is, but 
what it does, — namely, that it is something which 
gives to every particle of matter a tendency toward 
every other particle. This influence is conveyed 
from one body to another without any perceptible 
interval of time. We weigh a body by ascertaining 

Note. — It appears that in some if not all materials a limited amount of 
stress variation may be repeated time after time without apparent reduc 
tion in the strength of the piece ; on the balance wheel of a watch for 
instance, tension and compression succeed each other for some 150 mil- 
lions of times in a 3-ear, and the spring works for years without show- 
ing signs of deterioration. In such cases the stresses lie well within 
the elastic limits ; on the other hand the toughest bar breaks after a 
small number of bendings to and fro when these pass the elastic limits. 



the force required to hold it back, or to keep it from 
descending ; hence, also, weights are nothing more 
than measures of the force of gravity in different 
bodies. 

Inertia is that property of a body by virtue of 
which it tends to continue in the state of rest or 
motion in which it may be placed until acted on by 
some force. 

Kinematics. The science that treats of mo- 
tions, considered in themselves, or apart from their 
causes ; " Kinematics forms properly an introduc- 
tion to mechanics as involving the mathematical 
principles which are to be applied to its more prac- 
tical problems." 

LtOad. By the load on any member of a machine 
is meant the aggregate of all the external forces in 
action upon it. These may be distinguished as (i) 
the useful load, or the forces arising out of the use- 
ful power transmitted, and (2) the. prejudicial resist- 
ances due to friction, to work uselessly expended, to 
weight of members of the machine, to inertia due 
to changes in velocity of motion, and to special 
stresses caused in the apparatus by changes in its 
parts through variations of temperature. 



ROGERS' DRAWING AND DESIGN. 



809 



There are two kinds of load : first, a dead load 
which produces a permanent and unvarying amount 
of straining action, and is invariable during the life 
of the machine — such, for example, as its weight; 
and, second, variable or live load, which is alter- 
nately imposed and removed, and which produces 
a constantly varying amount of straining action. 
Everj' load which acts on a structure produces a 
chang-e of form, which is termed the strain due to the 
load. The strain may be either a vanishing or 
elastic deformation, that is, one which disappears 
when the load is removed ; or a permanent defor- 
mation or set, which remains after the load is re- 
moved. In general, machine parts must be so 
designed that, under the maximum straining action, 
there is no sensible permanent deformation. 

The Breaking Load is that load which causes 
in those fibres which are subjected to the greatest 
strain, a tension equal to the Modulus of Rupture ; 
in every case this is equal to the force necessary to 
tear, crush, shear, twist, break, or otherwise deform 
a body. 

Modulus. The primary signification of the 
Modulus is a measure ; the modulus of a machine 
means the same as the efficiency of it. " The modulus 
of a machine is a formula (or measure) expressing 



the work a given machine can perform under the 
condition under which it has been constructed"; 
the words mode, model, mold are kindred terms all 
formed from the same root-word and meaning some- 
what the same. 

Modulus of Resistance is the strain which corre- 
sponds to the limit of elasticity, compression and 
expansion each having a corresponding modulus. 
Modulus of Rtipture is the strain at which the mo- 
lecular fibres cease to hold together. Modulus of 
Elasticity is the measure of the elastic extension 
of a material, and is the force by which a prismatic 
body would be extended to its own length, sup- 
posing such extension were possible. The Modulus 
of a Machine is the amount of work actually ob- 
tained, divided by the work that should be obtained 
theoretically. 

Momentum means impetus or push ; it is the 
quantity of motion in a moving body ; it is always 
proportioned to the quantity of matter multiplied 
into the velocity. 

Moment is the tendency, or measure of tendency, 
to produce motion, especially motion about a fixed 
point or axis. 

Motion signifies movement ; in mechanics it may 
be either simple or compound, the latter consists of 



210 



ROGERS' DRAWING AND DESIGN. 



combinations of any of the simple motions. The 
acceleration of motion is the rate of change of the 
velocity of a moving body, in either an increasing 
or a decreasing rate. 

Power is the rate at which mechanical energy is 
exerted or mechanical work performed, as by a steam 
engine, an electric motor, etc. 

Theoretical Resistance is the force which, 
when applied to any body, either as tension, com- 
pression, torsion or flexture, will produce in those 
fibres which are strained to the greatest extent, a 
tension equal to the modulus of resistance ; or, in 
other words, it is a load which strains a load to its 
limit of elasticity. The Practical Resistance often 
improperly termed merely resistance, is a definite 
but arbitrary working strain to which a body may 
be subjected within the limits of elasticity. 

Ultimate Strength. If the straining action 
on a bar is gradually increased till the bar breaks, 
the load which produces fracture is called the ulti- 
mate or breaking strength of the bar. That ultimate 
strength is for different materials more or less 
roughly proportional to the elastic strength. We 
may insure the safety of a structure by taking care 
to multiply the actual straining action by a factor 



sufficiently large to allow, not only for unforeseen 
contingencies and the neglected causes of straining^ 
but also for the difference between the elastic and 
ultimate strength. The actual straining action mul- 
tiplied by this factor is still termed 2. factor of safety^ 
and is then equated to the ultimate strength of the 
structure ; the value of the factor of safety must be 
determined by practical experience. 

The Co-efficient of Safety is the ratio between 
the theoretical resistance and the actual load, or, 
what amounts to the same thing, the ratio between 
the elastic limit and the actual tension of the fibres. 
The Factor of Safety is the ratio between the 
breaking load and the actual load. 

As a general rule, for machine construction, the 
Co-efficient of Safety may be taken as double that 
which is used for construction subjected to statical 
forces. 

The Strength of Materials entering into ma- 
chine construction is measured by the resistance 
which they oppose to alteration of form, and ulti- 
mately to rupture, when subjected to force, pressure, 
load, stress or strain. 

Stress is the re-action or resistance of a body 
due to the load. 



ROGERS' DRAWING AND DESIGN. 



211 



Strain is the alteration in shape, as the result of 
the stress. 

Tenacity \?> the resistance which a body offers to 
being pulled asunder, and is measured by the tensile 
strength in lbs. per square inch of the cross section 
of the body. 

Tensile Strength is the resistance per unit of 
surface, which the molecular fibres oppose to separ- 
ation. 

Velocity is the rate of motion ; in kinematics, 
speed is sometimes used to denote the amount of 
velocity without regard to direction of motion, while 
velocity is not regarded as known unless both the 
direction and the amount are known. — (W. I. D.) 

Linear velocity is the rate of motion in a straight 
line, and is measured in feet per second, or per 
minute, or in miles per hours. Circjilar velocity is 
the rate at which a body describes an angle about a 
given point, and is measured in feet per second or 
per minute, or in number of revolutions per minute, 
as is a pulley or shaft. Uniform velocity takes 
place when the body moves over equal distances, in 
equal times. Variab 'e velocity takes place when a 
body moves with a constantly increasing or decreas- 



ing speed. Velocity ratio is the proportion between 
the movement of the power and the resistance, in 
the same interval of time. 

ViS-VlVa, or living force, is a term formerly used 
to denote the energy stored in a moving body ; the 
term is now practically obsolete, its place being 
taken by the word energy. 

WorU is the overcoming of a resistance through 
a certain space, and is measured by the amount of 
the resistance multiplied by the length of space 
through which it is overcome; the Principle of 
Work : The foot-pounds of work applied to a 
machine must equal the number of foot-pounds of 
work given up by the machine plus the number 
absorbed by friction. 



Note. — The simplest possible example of doing work is to raise a 
weight through a space against the resistance of the earth's attraction, 
that is to sa)', against the force of gravity. For instance, if a hundred 
pounds be raised vertically upwards, through a space of three feet, work 
is done, and, according to the above, the amount of work done is mea- 
sured by the resistance due to the attraction of the earth or gravity, i.e., 
one hundred pounds, multiplied by the space of three feet, through 
which it is lifted. The product formed by multiplying a pound by a 
foot is called a foot-pound. Thus, in the above instance, the amount of 
work done is 300 foot-pounds. Had the weight been only three pounds, 
but the height to which it was raised been 100 feet, the quantity of work 
done would have been precisely the same, i.e., 300 foot-pounds. 



212 



ROGERS' DRAWING AND DESIGN. 



PHYSICS. 

Physics is that branch of science which treats of 
the laws and properties of matter and the forces 
acting upon it ; especially that department of science 
(known, formerly, as Natural Philosophy) which 
treats of the causes that modify the general proper- 
ties of the bodies. 

The object of physics is the study of phenomena 
presented to us by bodies ; it should, however, be 
added that changes in the nature of the body itself, 
such as the decomposition of one body into others, 
are. phenomena, whose study forms the more imme- 
diate object of chemistry. 



MECHANICS. 

Mechanics is that section of natural philosophy or 
physics which treats of the action of forces on bodies. 

That part of mechanics which considers the action 
of forces in producing rest or equilibrium is called 
statics ; that which relates to such action in produc- 

NoTE. — " The mechanics of liquid bodies is also called hydrostatics 
or hydrodynatnics , according as the law of rest or of motion are con- 
sidered. The mechanics of gaseous bodies is called also pneumatics. 
The mechanics of 7?a?'rf5 in motion with special reference to the methods 
of obtaining from them useful results constitutes hydraulics." 

— Webster's International Dictionary. 



ing motion is called dynamics. The term mechanics 
includes the action of forces on all bodies, whether 
solid, liquid, or gaseous. It is usually, however, 
used of solid bodies only. Applied mechanics is the 
practical use of the laws of matter and motion in the 
construction of machines and structures of all kinds. 



PROPERTIES OF MATTER. 

The two essential properties of matter, both of 
which are inseparable from it, are extension and 
impenetrability. Extension, in the three dimensions 
of length, breadth, and thickness, belongs to matter 
under all circumstances ; and impenetrability, ox the 
property of excluding all other matter from the space 
which it occupies, appertains alike to the largest body 
and the smallest particle. 

The limits of useful knowledge relating to the 
properties of matter may be found in the three fol- 
lowing definitions : 

(a) " An atom is an ultimate indivisible particle 
of matter." 

(b) " An atom is an ultimate particle of matter 
not necessarily indivisible ; a molecule." 



ROGERS' DRAWING AND DESIGN. 



813 



(c) "An atom is a constituent particle of matter, 
or a molecule supposed to be made up of subordi- 
nate particles." — W. I. D. 

As no one really knows what matter is in the 
abstract, not even the most powerful microscope 
having- shown it, it were wise to rest here. 

The quantity of matter which a body contains is 
called its Mass ; the space it occupies, its Volume ; 
its relative quantity of matter under a given volume, 
its Density. All bodies have empty spaces denom- 
inated Pores. 

In solids, we may often see the pores with the 
naked eye, and almost always by the microscope ; in 
Huids, their existence can be proven by experiment ; 
there are reasons for believing that even in the 
densest bodies, the amount of solid matter is small 
compared with the empty spaces, hence it is inferred 
that the particles of matter touch each other only in 
a few points. 

There are also several other properties which are 
known by experience to belong to all matter, as 
gravity, inertia, and divisibility ; and others still 

Note. — The distinction between weight and moment is one impor- 
tant to have in mind. Weight, in mechanics, is the resistance against 
which a machine acts as opposed to the power which moves it ; moment, 
in mechanics, is the tendency or measure of tendency to produce motion, 
especially motion about a fixed point or axis. 



which belong not to matter universally, but only to 
certain classes of bodies, as elasticity, malleability, or 
the power of being extended into leaves or plates ; 
and ductility, or the power of being extended in 
length, as when drawn into wire. 

The mass of a body, or the quantity it contains is 
a constant quality, while the weight varies according 
to the variation in the force of gravity at different 
places. 



THE THREE STATES OF MATTER. 

Matter is any collection of substance existing by 
itself in a separate form ; matter appears to us in 
separate forms which however can all be reduced to 
three classes, namely, solids, liquids, gaseous ; a solid 
offers resistance to change of shape or shape of bulk, 
always keeping the same size or volume and the 
same shape ; a liquid is a body which offers no resist- 
ance to a change in shape and a gas or vapor is any 
substance in the elastic or air-like shape. 

Note. — The difference between a gas and a vapor is one less of kind 
than of degree. It is important to note that experiment proves that 
every vapor becomes a_gas at a sufficiently high temperature and low 
pressure, and, on the other hand, every gas becomes a \apor, at suffi- 
ciently low temperature and high pressure. 



214 



ROGERS' DRAWING AND DESIGN. 



THREE LAWS OF MOTION. 

As there are three states of matter already de- 
scribed, i. e., solids, liquids, gaseous, so there are three 
laws of motion. These are as follows : 

Law I. " Everybody continues in its state of rest, 
or of uniform motion in a straight line, except in so 
far as it is compelled by force to change that state." 

Law 2. "Change of (quantity of) motion is pro- 
portional to force, and takes place in the straight 
line in which the force acts." 

Law 3. " To every action there is always an equal 
and contrary reaction ; or the mutual actions of any 
two bodies are always equal and oppositely directed." 

The above are " Newton's Laws." 

Law one tells us what happens to a piece of matter left to itself, i. e., 
not acted on by forces; it preserves its " state," whether of rest or of 
uniform motion in a straight line. The first law gives us also a physi- 
cal definition of " time," and physical modes of measuring it. 

Law two tells us— among other things, how to find the one force 
which is equivalent, in its action, to anj* given set of forces. For, 
however many change of motion may be produced by the separate 
forces, they must obviously be capable of being compounded into a sin- 
gle change and we can calculate what force would produce that. 

Law three furnishes us with the means of studying directly the 
transference of energj* from one body or system to another. Experi- 
ment, however, was required to complete the application of the law. 



MATERIALS USED IN MACHINE CON- 
STRUCTION. 

The designer should not only know what provi- 
sions are to be made for strength, wear and tear, but 
he should also be familiar with the various mater- 
ials used in machine construction ; he should know 
what parts of the design are to be cast, forged, cast 
in one piece or framed or put together of many 
pieces and also how the work is done. 

The principal metals used in machine construc- 
tion are : Cast iron, wrought iron and steel. 

Cast iron is a mixture and combination of iron 
and carbon, with other substances in different pro- 
portions. The first smelting of the iron ore pro- 
duces pig iron. Pig iron is very seldom used in 
construction ; as a rule it is remelted and made into 
the kind of iron required for construction. The 
qualities of cast iron depend upon the proportion of 
carbon contained therein. 

There are different trades of cast iron : 

1st. White cast iron contains only a very small 
proportion of carbon ; it is very hard and brittle, it 
is mostly used for manufacturing wrought iron and 
steel. 

2nd. Gray cast iron contains part of the carbon in 
chemical combination and the rest is mechanically 



ROGERS' DRAWING AND DESIGN. 



215 



mixed with the iron in the form of graphite. Gray 
cast iron is divided into several kinds (mainly three), 
according to the quantity of carbon in the shape of 
graphite it contains ; these are Nos. i, 2 and 3. No. i 
contains the largest and No. 3 the smallest percent- 
age of graphite. The first kind has a great fluidity 
when melted and casts well; it has but little strength. 
The las.t kind has considerable strength and makes 
the mo.st rigid and massive castings. 

The great facility of casting this iron into any de- 
sired mold is the principal reason for its unlimited 
application and its great utility in machine construc- 
tion. 

It cannot be welded or riveted, it is very brittle 
and it has but little elasticity. 

These disadvantages cause the designer often- 
times to select more expensive materials. This iron 
is mostly used where rigidity and weight are of the 
utmost importance, as for instance in bed plates, 
frames, hangers, gears, pipes, etc. 

Wrought iron is produced by decreasing the 
quantity of carbon contained in cast iron ; it cannot 
be cast but can be worked into form by rolling or 
forging ; it can be welded, punched, riveted, etc., it 
is flexible and malleable. For shafts it is "cold 
rolled," thus adding to its strength and elasticity. 



Steel is refined or nearly pure iron, chemically 
combined with a certain per cent, of added carbon. 
Its great elasticity and strength make it the most 
suitable material for machine construction. Steel is 
divided into different varieties, according to the 
amount of carbon contained in it. Steel can be 
forged like wrought iron and it is fusible. Its hard- 
ness depends entirely upon the per cent, of carbon 
contained therein. According to its quality it may 
be used for cutlery, tools, springs and so forth. 

In selecting materials for machine construction, 
the most important properties that must be consid- 
ered, are : strength, stiffness, elasticity, weight, dura- 
bility, ease of manufacture and cost. 



MACHINES. 



Machines are divided into simple and compound; 
and machines when they act with great power, take 
the name, generally of engines, as the pumping en- 
gine. 

The simple machines are six in number, viz. : 

The lever. The wheel and axle. The pulley. 
The inclined plane. The screw. The wedge. 

These can in turn be reduced to three classes : 
I. A solid body turning on an axis. 2. A flexible 
cord. 3. A hard and smooth inclined surface. 



216 



ROGERS' DRAWING AND DESIGN. 



For the mechanism of the wheel and axle and of 
the pulley, merely combines the principle of the 
lever with the tension of the cords ; the properties 
of the screw depend entirely on those of the lever 
and the inclined plane ; and the case of the wedge 
is analogous to that of a body sustained between 
two inclined planes. 

All machines, however complicated they may be, 
are combinations of simple mechanical devices; the 
object in combining them is to give such a direction 

Note. — Man as a Machine. — " The human body forms an example of 
a machine. Physiologists calculate the work done by the body in foot 
tons, a foot ton of work being represented by the energy required to 
raise one ton weight one foot high. A hard-working man in his day's 
labor will develop power equal to about 3,000 foot tons, this amount 
representing both the innate work of his frame involved in the acts of 
living and his external muscular labor as a hewer of wood and a drawer 
of water. 

"A man's heart, in twenty-four hours, shows a return equal to 
120 foot tons ; that is, supposing he could concentrate all the work 
of the organ in that period into one big lift, it would be capable 
of raising 120 tons weight one foot high. The breathing muscles, in 
twenty-four hours, develop energy equal to about 21 foot tons, and 
when are added the actual work of the muscles and that expended in 
heat production 3,000 foot tons are arrived at as the approximate daily 
expenditure of energy. 

' ' All this power, moreover, is developed on about eight and one-third 
pounds of food a day, the supply including solid food, water and oxygen. 
No machine of man's invention approaches near to his own body, there- 
fore, as an economical energy producer ; and this for the practical rea- 
son that the human engine gets at its work directly and without loss of 
power entailed in other appliances that have to transmit energy through 
ways and means involving friction and other untoward conditions." 



and velocity to the motion as will enable the ma- 
chine to do the required work. 

The study of machines is divided by Reuleaux 
into the following parts : 

1. The study of machinery in general, looked at 
in connection with the work to be performed ; this 
teaches what machines exist and how they are con- 
stituted. 

2. The theory of machines, which concerns, itself 
with the nature of the various arrangements by 
means of which natural forces can best be applied 
to machinery. 

3. The study of machine design, the province of 
which is to teach how to give the bodies constituting 
the machine the capacity to resist alterations of 
form. 

4. T+ie study of pure mechanism, or of kinematics, 
which relates to the arrangements of the machine by 
which the mutual motions of its parts, considered as 
changes of position, are determined. 

Upon these foundation principles have been con- 
structed many thousands of machines ; instances are 
on record where the number of tools and machines 
have run into the tens of thousands used in a single 
shop, in another more than three thousand "jigs" 
were in use ; from this may be perceived the possi- 
bility of describing but few of the many examples. 



ROGERS' DRAWING AND DESIGN. 



217 



STRESSES, STRAINS AND LOADS. 

The great variety of materials employed in ma- 
chine construction precludes a complete table of 
factors of safety for use in practice for various ma- 
terials under dead and live loads and for machines 
subjected to sudden and frequent strains of short 
duration, known as shocks. 

We give here only a few of the most important 
materials showing the factors allowed in general 
practice : 

FACTORS OF SAFETY. 

Material 
MATERIALS. Varying subject to 

Dead Load. Load. shocks. 

Cast Iron 6 10-15 15-20 

Wrought Iron 4 6 12 

Steel 4 7 15 

Copper 5 8-10 10-15 

Timber 8 10 15 

Masonry and Brickwork 15 25 30 

The stresses to which constructions and parts of 
constructions may be subjected are of three kinds, 
mainly • 

I. Tensile strain or stress, which has a ten- 
dency to lengthen the body in the direction of the 
load. 



2. Compressive or crushing strain or stress, 
which produces a tendency to shorten or crush the 
body in the direction of the load. 

3. Shearing strain or stress produced in a piece 
of material which is distorted by a load, tending to 
cut it across. 

Various metals have a different strength to resist 
compressive and tensional stresses. Stress is usually 
measured in pounds per square inch. 

As mentioned above, when a part is not loaded 
beyond its limit of elasticity, the stress produced is 
directly proportional to the strain, so that the 
stress divided by the strain is a constant quantity 
for the same material. This constant quantity is 
called the modulus of elasticity. The modulus of 
elasticity is found by dividing the stress by the strain. 

The modulus of elasticity is also called the co- 
efificient of elasticity. 

If a cross section of a given bar is equal to A 
square inches and if this bar is subjected to a load 
of W pounds which may result in tensile or com- 
pressive stresses, and if the modulus of elasticity 
of the material in the given bar is equal to E 
pounds per square inch, then the strain produced is 
determined by the formula : 

W 



Strain = 



E X A 



218 



ROGERS' DRAWING AND DESIGN. 



MODULUS OF ELASTICITY. 
Materials. Pounds per sq. in. 

Cast Iron 1 8,000,000 

Wrought Iron (in bars) 29,000,000 

" " (in plates) 26,000,000 

Steel 30,000,000 

Brass (cast) 9,000,000 

" (wire) 1 4,000,000 

Copper (in sheets) 15,000,000 

" (wire) 1 7,500,000 

The stress or load per sq. in. of section is called 
the unit stress. For instance, if a bar is subjected 
to a load of 2,000 lbs. and the cross section of the 
bar is equal to 4 sq. in., the unit stress of the bar 
would be 2000 ^4=500 lbs. 

As has been said before, strain is the amount of 
alteration in form of a piece of material produced 
by a stress to which the piece is subjected. If 
a wrought iron bar is subjected to pulling stress 
and is, as a result of this, lengthened titVt of an 
inch, this change in its form in length or area, as 
may be the case, is called the strain. 

The unit strain is the amount of alteration of 
form per unit of form. It is usually taken per unit 
of length, and then it is called the elongation per 
unit of length. 



We may express this in the following formula : 



Strain = 



increase in length of bar 
original length of bar 



For instance if a bar 8 ft. long is elongated by jV 
of an inch when subjected to a load, the strain is 
equal to tV divided by 96 



rsiTT- 



It is to be remembered that the relation of the 
proportion between the stress and the strain ip true 
only within the elastic limit. 

The smallest load which will cause the rupture of 
a piece of material is called the ultimate strength of 
that piece, that is the stress in lbs, per sq. in. which 
the piece can sustain just before rupture takes 
place. 

The following is a table of ultimate strengths : 

ULTIMATE STRENGTH IN POUNDS PER SQUARE INCH. 
Material. Tensile. Compressive. Shearing. 

Cast Iron 19,000 90,000 20,000 

Wrought Iron... 52,000 52,000 50,000 

Steel 100,000 1 50,000 70,000 

Wood 10,000 8,000 600 to 3,000 



I 



ROGERS' DRAWING AND DESIGN. 



219 



Example i, Fig. 303. 



A wrought iron bar 2" by 2" in section is subjected 
to tension by the action of a load ; it is required to 
find the weight which zvill cause its rupture. 

The foregoing table of ultimate 
strengths shows 52,000 lbs. per sq. in. 
as the tensile stress, and as the given 
bar measures 4 sq. in., in section, the 
load required is 52,000x4^208,000 
lbs 





Fig. 308. 



Fio. aM. 



Example 2, Fig. 304. 

A square cast iron block is required to sustain a 
load 0/80,000 lbs. What must be the Icjigth of a 
side of this block ? 



Let us employ a factor of safety, say 5 for cast 
iron. That is, we will suppose that the load which 
is to be sustained will be 5 times greater than 
80,000, i. e., 400,000 lbs. As the piece of material 
in question is subject to a compressive stress, we 
find the ultimate strength of cast iron in compres- 
sion, go,ooo per sq. in. of section. To find the re- 
quired area of a section, divide the load, 400,000, 
by the ultimate strength, 90,000, -Vir°iW-= 4-444 
square inches. This is the square section required 
for the block ; so to find the length of a side, take 
the square root of 4.444 which is 2.1081 inches, 
or about 2^^ inches. 

From this example, as well as from what has been 
said above, we draw the following conclusions : 

The resistance to compression, of a piece which 
is short, compared with its cross section, is calcu- 
lated by the following formula : 

Load = Area of Section x Compressive Stress. 

The compressive strength of materials is gener- 
ally much more difficult to determine when the ma- 
terial is of a soft and plastic character which causes 
them to spread out when under compression. 

The method here described for the calculation of 
the compressive strength of materials is true only 
in the case where the given piece is comparatively 
short. Longer pieces of material, subjected to com- 



220 



ROGERS' DRAWING AND DESIGN. 



pressive stresses are much more difficult to calculate 
because of other strains arising from the action of 
the load. 

The resistance to tension is calculated as in the 
first example. If a piece having a cross section 
of A square inches is subjected to a tensile stress 
by the action of a load of W pounds, and if the ten- 
sile strength in pounds per sq. in., is uniformly dis- 
tributed over the cross section, and is equal to f, 
then the load W = A (area) x f (slre"ngt'h). 

Resistance to shearing is calculated by the same 
formula as the resistance to tension, namely : 

W = A X f . 

The ultimate shearing strength of metals is 
usually from 70 to 100 per cent, of their ultimate 
tensile strength. 

Stresses induced by bending. 

When a beam is supported at both ends, the load 
causes the material, in the upper part, to be com- 
pressed and that in the lower part to be stretched. 

We may imagine a horizontal surface separating 
the compressed part of the beam from the stretched 
part. We shall call this surface the neutral surface 
of the beam. The line in which this surface inter- 
sects a transverse section of the beam is called the 
neutral axis of that section. 



It is evident that a beam may have as many neu- 
tral axes as there are cross sections taken in the 
beam. The bending stresses occurrino- in a beam 
supported at both ends will depend not only upon 
the magnitude of the forces acting thereon, but also 
on the distances of the line of action of the given 
forces from any section of the beam under consider- 
ation. 

At any point in the length of the beam, the bend- 
ing action is equal to the sum of all external forces 
at that point. This is expressed generally as follows : 
the bending action must be measured by the moments 
of the forces acting on the beam relative to the 



given section. 



The moment of a force is equal to the force 
multiplied by the length of the perpendicular to the 
direction of the force, from a point in which the 
beam is supposed to be fixed. In Fig. 305 the 
moment of the force induced by the weight of 20 lbs. 
is equal to 20 times 10 ^= 200 ft. pounds. 

In Fig. 306, the moment is equal to 20 x 9= 180 
ft lbs. 

The resultant moment of the forces acting on 
the beam on one side of a given section, referred 
to that section, is called the bending moment on 
the beam at that section. For instance, in a beam 
fixed at one end and loaded at the other with a 



ROGERS' DRAWING AND DESIGN. 



221 




Fig. 306- 



K Jft 



f 





Fig. 307. 



weight of lOO pounds, Fig. 307, the bending moment 
at a cross section at a distance of 5 ft. from the free 
end of the beam is 100 x 5=500 ft. lbs. 

In Fig. 308, a beam supported at both ends is 
shown, and where a uniformly distributed load of 
W pounds per unit of length and a concentrated load 
of W pounds at a distance a from one end is given, 



n 



R'^ 




-X 



Y 
iVJC 



-a- 



-> 



2 



w 



Fig. 308. 



let it be required to find the bending moment at g 
section, a distance x from one end. First deter- 
mine by the principle of the lever, the reactions R 
and R' of the points of support. The forces to the 
left of the given section are R, W and w X x. The 
moments of these forces relative to the section are 



222 



ROGERS' DRAWING AND DESIGN. 



R X X, W X (x - a) and -^ and the 
resultant moment Rx-W X (x-a) - ^^ 
and this is the required bending mo- 
ment at the given section. 

The combined compressive stresses 
on one side of the neutral axis of any 
cross section are equal to the com- 
bined tensile stresses on the other 
side of that axis. These two equal 
and parallel forces form a couple, 
whose moment is the moment of 
resistance of the beam to bending- of 
that section. The moment of resist- 
ance is equal to the bending mo- 
ment. 

Suppose that the greatest com- 
pressive or tensile stress at a given 
section of a beam is equal to f, then 
we may express the moment of resist- 
ance by the product of fz, where z 
is a quantity called the modulus of 
the section, depending upon the form 
of the section of material in consid- 
eration. 

The modulus of section or section 
modulus is sometimes called resisting 
inches of a section. 



Form of 
Section. 




^--B--^ 




i^t- 



Fig. 311. 




Section 

MODULU.S. 



i }3D' 



I'll:. ;ji('i. 



fJB' 



>, o.//sB^ 



9) is. 



•3./W6 j^., 



Fig. 312. 




Section 
Modulus. 



_. 3.W6 r>^. 



6^ n / 



..lJ ^ ^^ D J 



Fiu. S].5. 






i 



-i. 



''■"■3- 



Fj(i. 31B. 



ROGERS' DRAWING AND DESIGN. 



223 



Form of 

Section. 



Section 
Modulus. 




Kii;. :11T. 




Fig. 318. 







Fig. :ilH. 





Section 
Modulus. 






:♦ - d- -)i 



\X J- f6<idtL7i^) 



X 



Fig. 3«). 







^ Jzl—. — J 



The relation between the 
bending moment and the mo- 
ment of resistance may be ex- 
pressed by the formula M=fz. 

It must be remembered that 
this ratio is only true as long as 
the elastic limit of the beam 
has not been reached. 

The formula is based on the 
supposition that the stress being 
greatest at the top or bottom 
of a cross section, diminishes 
gradually to nothing at the neu- 
tral axis of the section. 

The following illustrations, 
Figs. 309-324, give the value of 
the section modulus z for various 
sections. The horizontal line 
drawn in each section represents 
the neutral axis. 

The safe resisting moment is 
equal to the safe stress of the 
material multiplied by the sec- 
tion modulus. 



224 



ROGERS' DRAWING AND DESIGN. 



Example 3. 

What is the safe resisting moment of a wooden 
beam, the extreme fibre stress of which is equal to 
6,000 lbs. per sq. in., when the beam is lo" wide and 
1 2" deep ? 

Take a factor of safety equal to 6. 

The section modulus for the given section is, 
according to the table ^21 = '"^^'t" = 240. 



B and is equal to W X L, the weight of the load 
multiplied by the length of the beam. 

There also exists a shearing force equal to the 
magnitude of the load W, which force is to be taken 
equal from end to end of tlie beam. 

Example 5. 

In Fig. 326 is shown a beam fixed at one end, 
carrying a load W uniformly distributed. The 





Ftg. 325. 

The extreme stress for the given material is equal 
to 6,000 lbs. As a factor of safety equal to 6 has 
been taken into account, we divide 6,000 by the 
factor 6^1,000 lbs. 

Then the safe resisting moment of a section of the 
beam is equal to 240 x 1,000=240,000 inch pounds. 

Useful examples of bending moments. 

Example 4, Fig. 325. 

The beam is fixed at one end while the load acts 
at the other. The greatest bending- moment is at 



vmrn 




w 

A 




^: \ V 


^• 


- 


\ 


y//////A//// 


,, J. 


i 


B 


mwM 




. T 


-> 


fy/Mm^ 




Lt 


m^r<i< 



Fig. 326. 



greatest bending moment is again at B and is equal 
to % WL. 

Example 6. 

In Fig. 327 we have a beam supported at each 
end and loaded exactly in the middle. The great- 
est bending moment in this case is at B and it is 
equal to ^ W X L. There also exists a shearing 
force equal to ^ W, this force being uniform 
throughout. 

o 



ROGERS" DRAWING AND DESIGN. 



225 



The beam shown in Fig. 328 is supported at each 
end and is loaded uniformly. The greatest bending 
moment is at the middle of the beam and is equal 
to 5^ W X L- When compared with the bending 



The greatest shearing force is at the ends near the 
supports and is equal to yi W. 

Example 8. 

When the beam is fixed securely at each end, and 





B 



---it- 



FiG. 32T. 





4 .'If T y 4 '1 




Fig. 328. 



moment in the preceding example, we see that a 
beam may carry a load two times greater when the 
load is distributed uniformly throughout the beam. 



loaded at the center, as in Fig. 329, the greatest 
bending moment is at the center and at each end, 
and is equal to i/^ W x L- This is based on the 



226 



ROGERS' DRAWING AND DESIGN. 




Fig. 329. 



supposition that the cross section of the beam is 
uniform throughout its full length. The bending 
moments at the ends are contrary to the bending 
moments at the center, that is, at the center, the 
bottom of the beam 
will be subjected to \y//y////A ^^\ 

tension, while at the |^^^ . . | . — j — |>V 

ends the bottom f///^M(^ \\\\\\\\ 

will be subject to 

compression. 

Example 9, Fig. 

330. 

When a beam, 
having a uniform 
cross section from 
end to end is fixed 
securely at both 
ends, the load which 
the beam is made to 
carry, being distrib- 
uted uniformly, as 
in Fig. 330, the bending moment is greatest at the 
ends and is equal to tV WL. The bending moment 
at the center is equal to yi of the moment at the 
ends, that is, equal to 5V WL, and is contrary to the 
moments at the ends. 

If a beam is required to be very stiff, the length 





Flu. ;J30. 



should be made as short as possible and the depth 
as great as circumstances will permit. With the same 
area of section, the deeper the beam the stronger 
it will be, provided the breadth of the beam is suf- 
ficient to prevent 
'y///X^///^^ lateral breaking. 

Various applica- 
tion of the princi- 
ples of strength of 
materials will' be 
discussed in connec- 
tion with the design 
of different parts of 
machines. 

Another requisite 
for successful de- 
signing is a knowl- 
edge of the proper- 
ties of materials 
commonly used for 
machine construc- 
tion. In selecting materials for machine parts the 
designer must consider their properties in regard 
to the adaptability for the work to which they 
are to be subjected ; the strength, stiffness, dura- 
bility and convenience of working into the necessary 
form. 




ROGERS' DRAWING AND DESIGN. 



2Ji7 



A machine properly constructed, must be able to 
withstand the stress to which its various parts are to 
be put, and this depends entirely on their action and 
endurance, as conditioned by the forms of the parts 
of the machines. By the word stress we mean a force 
acting between two bodies or two parts of the same 
body when subjected to the action of a load. This 
force is understood to resist the load in preventing it 
from changing the form of the machine or its parts. 

The combination of all external forces acting on 
a part of a structure calls into e.xistence a new force 
within the structure itself, and this resisting force 
we call stress. 

All the external forces are called the load of the 
machine. The effect of the load is the strain pro- 
duced in the machine ; the strain is the tendency to 
change the form of the machine part under the 
influence of the load. The resistance which is 
offered by a material to the change of form resulting 
from the application of a load, combined with its 
natural power of returning to its original shape after 
the load is removed, is called its elasticity. 

When a piece of material deformed somewhat 
when subjected to a load returns exactly to its 
original form as soon as the load is removed, the 
piece of material is said to be perfectly elastic with- 
in certain limits of a load. 



When under the influence of a load the piece of 
material is permanently deformed — that is, does not 
return to its original form when the load is removed 
— we say that the limit of elasticity of the material 
has been reached. 

Up to the limit of elasticity the stress is directly 
proportional to the strain ; beyond the limit of 
elasticity the strain increases taster than the stress 
until rupture is produced. 

The loads to which material can be subjected may 
be divided primarily into two classes : a dead load 
is one which is applied slowly and remains steady 
and unchangeable ; a live load is one which con- 
stantly changes, being either alternately imposed and 
removed, or varying in intensity and direction. 

To avoid the danger accompanying an unforeseen 
intensity of strain, which may produce undesirable 
deformation or rupture, as may be caused by imper- 
fect workmanship, poor quality of material or other 
causes, the parts of a machine are usually made to 
resist a much greater load than will be brought on 
them in the regular course. The expected load is 
supposed to be greater, and for this reason is multi- 
plied by a number known as the factor of safety. 
The factor of safety varies for different materials 
according to their structure and application, as well 



228 



ROGERS' DRAWING AND DESIGN. 



as for the same kind of material according to con- 
ditions to which it may be subjected. 

For materials, the quality of which is liable to 
change, the factor of safety must be larger than for 
materials the quality of which is more uniform and 
less liable to change through atmospheric exposure 
or varying temperature. 

It happens that in some structures the whole load 
cannot be ascertained with accuracy — in such cases 
the factor of safety must be increased as a safeguard 
against unexpected straining action. It may also 
happen that in some machines the working load may 
be suddenly increased — for such accidental strains a 
factor of safety must be allowed. 



SCREWS, BOLTS AND NUTS. 

In all working drawings consideration should be 
given to the manner of uniting the different parts of 
the machine. Screws play a most important part in 
machine design, particularly as a means for fastening 
the different parts together. The representation of 
bolts and screw threads is consequently of such im- 
portance that a knowledge of their proportions and 
the usual method of drawing them, is of great con- 
sideration to machine draughtsmen ; the exact repre- 
sentation of a screw thread is somewhat difificult ; it 
takes both time and care. 



The proper way to draw a screw thread as it ac- 
tually appears in a finished screw, is by laying out a 
curve or curves upon the surface of the cylinder, 
forming the body of the bolt. This curve is called 
a helix ; the helix may be defined as a curve gener- 
ated by progressive rotation of a point around an 
axis, remaining equidistant from the axis through- 
out the length of the motion. 

When a machinist desires to cut a thread upon a 
cylinder, he will first change the gears of the Lathe 
to produce the desired number of threads toreach 
inch of length of the screw ; this being done, the 
cylinder is put in place on the centers of the lathe 
and the thread cutting tool is then set to its proper 
angle. 

Before proceeding to cut the thread, the tool is 
moved close to tiic work, so as to trace a fine line 
upon the surface of the cylinder when the machine 
is put in motion ; the fine spiral produced upon the 
surface of the cylinder in this manner, is the helix of 
the screw. 

Problem : To dram a helix, the diameter and 
height of one turn being given. 

The heiijht of one turn of a helix is called its 
pitch. 

Let the diameter of the cylinder be 3" and the 
pitch 2". 




ROGERS' DRAWING AND DESIGN. 



229 



Draw the elevation of the cylinder ABCD above 
its bottom view i, 2, 3, etc., Fig. 331. The eleva- 
tion ABCD may be four inches high, that is equal 
to two complete turns of the helix. 

Lay off the pitch from the point A upon the line 
AB equal to A 12 and 12 B. Divide the pitch A 12 
into any number of equal parts, for instance in this 
case 12. Divide the circle into the same number of 
equal parts. Through the points of division on the 
circle, draw lines parallel to the line AB and ex- 
tend them through the full height of the front view 
ABCD. 

Through the point i of the divisions of the pitch, 
draw i-i' parallel to AD, intersecting the vertical 
line I i' in the point i' which is a point in the re- 
quired helix. Through the point 2 of the pitch divi- 
sions, draw the line 2-2' parallel to AD and inter- 
secting the vertical line 2 2' at the point 2', which is 
another point of the helix. Through the point 3 of 
the pitch divisions draw the line 3-3' parallel to AD 
and cutting the vertical line 3 3' in the point 3', 
which is a third point of the required helix. Pro- 
ceed in this manner until the sixth point of the helix 
is found : it will be situated on the line DC. The 
points A, i', 2', 3', 4', 5' and 6' determine the position 
of one-half of a turn of the helix, which may be traced 
through these points, first in pencil and then inked in. 



330 



ROGERS' DRAWING AND DESIGN. 



In the same mariner the second half of the first 
turn may be completed. The accompanying illus- 
tration renders a repetition of the above explana- 
tion unnecessary. The second half of the turn is 
drawn dotted, as it is on the other side of the cylin- 
der and cannot'be seen. The second turn may be 
laid out by the aid of the points of the first turn of 
the helix in the following manner. Set the com- 
passes to a distance equal to the pitch and lay ofi 
the points i", 2", 3", etc., above the corresponding 
point i!, 2', 3', etc., of the first turn of the helix. 

A thorough understanding of the above problem is 
of considerable use, not only for drawing large sized 
screws, but especially for drawing a worm for worm 
gears, which will be explained later. 

A screw with a V-thread, drawn with exact helical 
curves is shown in Fig. 332. 

It is made of two helices, one for the top of the 
thread and the other for the root of it. 

Fig. 333 shows a screw with a square thread. An 
examination of the drawing will show that the thread 
is drawn with four helices ; two helices upon the 
outside of the cylinder, the top of the thread, and 
two for the root of the thread. It is evident that 
the method of drawing screw threads with helices 
while producing an exact representation of the screw 



cannot be employed in the shop in drawing machin- 
ery, where, as a rule, the number of bolts and screws 
is very considerable. 

The bolts, nuts, etc., are so numerous on some 
machines, that it is customary to make separate dol^ 
sheets, showing all screws necessary for one machine, 
in all their different sizes and forms. 

The square thread shown in Fig. 333, would 
appear, when drawn by straight lines only, as in Fig. 
334, and the V-thread shown before would be drawn 
as in Fig. 335. v^ 

We have so far considered only right-handed 
screws. A right-handed screw is one, which passing 
through a fixed nut and turned in the direction of 
the motion of the hands of a clock, will advance into 
the nut. 

A left-handed screw is one, which to pass through 
a fixei nut^ must be turned in a direction opposite to 
the motion of the hands of a clock. Such a thread is 
shown in Figs. 336 and 337. Screws maybe either 
single-threaded or double-threaded. If we assume 
that a screw consists of a cylinder with a coil form- 
ing the thread wound around it, we may easily define 
a double screw as a cylinder with two parallel coils 

Note. — To avoid the difficult and tedious operation of drawing the 
helices, screw threads are generally indicated by straight lines only. 



ROGERS' DRAWING AND DESIGN. 



231 






Fig. 333. 



Fig. 33:3. 



Fig. 334. 



232 



ROGERS" DRAWING AND DESIGN. 



of thread wound around it. Generally the double- 
threaded screw is defined as one having two paralled 
threads. A screw having three parallel threads is 
called a triple-threaded screw. Double-threaded 
screws are shown in Figs. 338 and 339. 

The distance betzveen the centers of two successive 
threads in a single-threaded screw is called the pitch 
of the screw. Figs. 340 and 341 ; the pitch is equal 
to the distance tvhich the screw will advance into a 
fixed nut during one turn. 

Fig 341 shows the pitch of a square thread. It is 
equal to twice the pitch of the triangular thread. 
Screw threads are generally either triangular or 
square in section, although some other forms are in 
use. 

The triangular thread is called the V-thread. The 
form of V-thread most commonly used in this country, 
known as the U. S. Standard thread, is shown in 
Fig. 342. 

The U. S. Standard screw, known also as the 
Franklin Institute Standard, was presented to that 
Institute by Mr. Wm. Sellers, in a paper read by 
him in 1864. As a result of this, the Franklin Insti- 
tute recommended for general adoption by American 
engineers the following rules and table of standard 
threads : 



Table of U. S. Standard Screw Threads. 



(Outside) 

Diameter of 

Screw. 


Threads per 
Inch. 


Diameter at Root 
OF Thread. 


Diameter op 
Tap Drill. 


\ 


20 


0.1S5 


-h 


A 


18 


0.240 


\ 


3 


1(5 


0.294 


A 


t\ 


14 


0.344 


S3 
7T 


\ 


13 


0.4011 


il- 


-h 


12 


0.454 


¥1 


% 


11 


0.507 


rf 


\ 


10 


0.620 


1" 


1 


9 


0.731 


1 


1 


8 


0.837 


M 


H 


7 


0.940 


M 


li 


7 


1.065 


1 3 


If 


6 


1.160 


lA 


u 


6 


1.284 


I5V 


If 


5i 


1.389 


m 


If 


5 


1.491 


n 


H 


5 


1.616 


i| 


2 


4i 


1.712 


i|- 


2i 


# 


1.962 


2 


^ 


4 


2.176 


'H^ 


2| 


4 


2.426 


h\ 


3 


3i 


2.629 


2p- 


H 


3i 


3.100 


H 


4 


3 


3.567 


H 



ROGERS' DRAWING AND DESIGN. 



233 



The proportion of pitch to diameter is 
P=o.24 -v/D + 0.625 — o. 1 75 
The depth of the thread is 0.65 of the pitch. 



The table does not give the pitch. To find the 
pitch, divide one inch by the number of threads. 
Eight threads to one inch give a pitch of ys". 






FlO. 3%. 



FlO. 336. 



FlO. 337. 



By the term diameter of the screw is always meant 
the outside diameter. The diameter measured at 
the root of the thread is called the inside diameter. 



In the foregoing table of U. S. Standard Screw 
Threads, the number of threads to one inch of screw 
to 4" in diameter. 



is given from i^ 



234 



ROGERS' DRAWING AND DESIGN. 



The third column gives the diameter of the screw 
at the root, or the inside diameter. The next column 
gives the diameter of drill to be used for any required 
diameter of tap or thread. They are ordinarily a little 
laro-er than the diameter at the root of the thread. 




Fig. 338. . 

The screw thread is formed with straight lines at 
an angle of 60° to each other. The top and bottom 
of the thread are flattened, each to a width of ^^th 
of the pitch, Fig. 343. 

For small diameters of bolts the amount of flat- 
tening is not made to any particular measure, and in 



drawing screw threads it may be neglected entirely. 
For a square-threaded screw, the number of threads 
per inch is equal to one-half the number on a V- 
threaded screw. 




Fig. 339. 

In a square-threaded screw of U. S. Standard 
form, the width of the thread is equal to the width 
of the groove — each equal to one-half the pitch, 

Fig- 344- 

The depth of the thread is also equal to one-half 
of the pitch — that is, equal to the width of the groove. 



ROGERS' DRAWING AND DESIGN. 



235 



Figs. 345-350 exhibit the conventional methods 
of showing; different threads of a bolt. 




Figs. 340 and 341. 



Fig. 346 represents a single square-thr aded screw. 
To draw the screw, first draw the cylinder. Lay off 
distances each equal to one-half the pitch and through 
the division points draw lines at right angles to the 
axis of the cylinder, and cutting the other side of 
the cylinder, the inclination of the parallel lines in- 
dicating the thread through the width of the cylinder 
being equal to one-half the pitch. This method is 
clearly illustrated in Fig. 346. 




To 



Fig. 312. 

Fig. 345 shows a single V-threaded screw, 
draw the screw lay out the outlines of the cylinder 
of the bolt and upon one of its sides set off distances 
each equal to the pitch. Do the same on the other 
side of the cylinder beginning at a point one-half the 
pitch from the end of the cylinder, after which draw 
the lines for the top of the thread. From the points 
of division draw lines inclined to each other 60° for 



236 



ROGERS' DRAWING AND DESIGN. 




FiQ. 343. 



Pitch i 



1^- i Pitch >k • -k P'fcf* 




Fig. 3«. 



the threads, on both sides of the cylinder, then con- 
nect the roots by straight lines. It will be noticed 
that these lines are not parallel to the lines connect- 
ing the tops of the thread. 

Fig. 347 shows a still simpler method of represent- 
ing a V-threaded screw. The pitch is laid oft as in 
the preceding example. The heavy lines represent 
the bottom of the thread. 

The method employed in Fig. 348 is still more 
rapid in delineation and is, therefore, recommended 
for rapid drawing. Here the heavy lines are used 
to represent the top of the thread, the fine lines in- 
dicatinjj the bottom of the thread. 

In Fig. 349 the fine lines are drawn as long as 
the heavy lines, which makes the drawing of the 
thread still easier. A method of indicating screw 
threads when great haste is necessary and for sketch- 
ing is shown in Fig. 350. 

In drawing the thread as illustrated in the last 
four figures, no particular attention need be given to 
the number of threads per inch. A note written 
plainly on the drawing, very near to the representa- 
tion of the screw, gives the exact number of threads 
to the inch. Even this may be left out when the 
diameter of the screw is plainly given, with the note 
"standard" near it; in this case the workman is 
expected to determine the number of threads to 
the inch from the table of U. S. Standard Threads. 



ROGERS' DRAWING AND DESIGN. 



237 



The proportions of bolt heads and nuts which 
have been accepted in this country as a standard are 
as follows : 

The distance between the paral- 
lel sides of heads and nuts is equal 
to I Yi times the diameter of the 
bolt, plus y% inch=i3^ D + 5^ 
inch. 

The thickness of heads is equal 
to one-half of the distance between 
the parallel sides. 

i}4 D + ys inch. 
2 

The thickness of nuts is equal to 
the diameter of the bolt^ D. 

The same proportions are used 
for square heads and nuts. 

In all these formula; D expresses 
the diameter of the bolt. 

Fig. 351 shows the conventional 
method of representing a hexagonal 
nut for a 2" bolt, The height of the 
nut is equal to 2". The two views 
may be drawn similar to the two 
views of a hexagonal prism, ex- 
plained in the chapter on projec- 
tion. 



The curve cde is drawn first, with a radius equal 
to the height of the nut. When the points c and e 
are thus determined, a fine straight line is drawn 




Fia. :i45. 




Fia. 346, 






Fio. 349. 




238 



ROGERS' DRAWING AND DESIGN. 




Fig. 351. 



ROGERS' DRAWING AND DESIGN. 



239 



through these points and extended in both direc- 
tions so as to cut all vertical edges of the nut in both 
views, at the points a, g, h, k and m. Arcs are then 




Fig. 352. 



Fig. 353. 



drawn through the points a, b, c and through e, f, g. 
The same is done in the other view in passing arcs 
through h, i, k and k, 1 and m. These arcs are struck 



with compasses, after a centre is found by trial with 
the compasses. The chamfer at aa and g3 may be 
drawn by 45° lines, from the points a and g respec- 
tively. 





Fia. 354. 



Fig. a55. 



This is not the exact construction of the curves as 
they appear on a hexagonal nut. However, the 
exact curves are not of any importance on a work- 
ing drawing, and it will be found that this prac- 



240 



ROGERS' DRAWING AND DESIGN. 



tical shop method effects a material saving of time 
and trouble, particularly as the representation of 
heads and nuts is of very frequent occurrence in ma- 
chine drawing. In drawing a hexagonal nut or head, 
it is the general custom to show three faces of each. 

A square nut or bolt head is 
generally shown by drawing one 
face of each only. Fig. 352 illus- 
trates a bolt with hexagonal 
check nuts. It is more conven- 
ient to make both nuts of stand- 
ard thickness, that is equal to the 
diameter of the bolt, although it 
is often found that the inner nut 



is made thinner. In the illustration the outer nut 
is chamfered on both faces. 

In Fig. 353 a bolt with a square head is shown. 

The distances between the parallel faces of this head 

is equal to i y^ times the diameter of the bolt plus 

y% inch. The height is equal 



the distance between the 



to ^ 

parallel faces. The arc for the 
chamfer of the head is usually 
drawn with a radius equal to 2^ 
times the diameter of the bolt. 

A set screw is shown in Fig. 
354. The figure illustrates all 
required proportions, as they are 




Fig. 356. 



ROGERS' DRAWING AND DESIGN. 



241 



commonly used. The point of the set screw is 
usually made with an arc having a radius equal to 
four times the diameter of the screw. 

A stud-bolt is one which is threaded at both ends. 
Fig. 355, one end being screwed into one of the 





i... 



'rd- 



-lU- 






— I--- 




FiQ. 3oT. Fig. 358. Fig. 3.59. 

pieces of a machine to be connected, while the other 
end passing through the other piece, which is to be 
fastened to the first, carries an ordinary nut, as in 
Fig. 356, which illustrates how a stuffing box is 
fastened to a cylinder head. 



The conventional way of representing screws with 
square heads is shown in Figs, 357, 358. A round 
head screw is shown in Fig. 359. The head of the 
screw is slotted. In the top view the parallel lines 
showing the slots should be drawn at an angle of 
45^^ with a horizonal line. This head is particularly 
adopted for countersunk work. 

In conclusion a few words are added concerning 
the strength of bolts. Tke weakest part of the bolt 
is the section at the bottom of the thread. The fol- 
lowing is a table of the tensile strength of U. S. 
Standard Bolts at 5,000 lbs. per sq. in. : 

Tensile Strength of U. S. Standard Bolts 
AT 5,000 LBS. PER Square Inch. 



Diameter of 


Tensile 


Diameter of 


Tensile 


Screw. 


Strength. 


Screw. 


Strength. 


\ 


134 


1-1 


5,300 


■h 


226 


H 


6,400 


1 


339 


H 


7,650 


1^ 


465 


If 


8,800 


i 


625 


i| 


10,150 


■^ 


809 


2 


11,500 


% 


980 


H 


15,600 


\ 


1,500 


H 


18,500 


\ 


2,100 


4 


23,000 


1 


2,750 


3 


27,200 


H 


3,450 


3^ 


37,700 


li 


3,900 


4 


49,500 



The figures in the second and fourth columns show the total load 
which can be sustained hy bolts of the above diameters. In calculating 
the strength of a bolt the stress to which it is subjected by the use of 
the wrench must be taken. 



242 



ROGERS' DRAWING AND DESIGN. 



The figures in the second and the fourth columns 
show the total load which can be sustained by bolts 
of the respective diameters. In calculating the 
strength of a bolt, the stress to which it is subjected 



resisting strength, the value of the safe stress per 
square inch of section must be taken comparatively 
low, and it is advisable for the purpose of overcom- 
ing all difficulties here mentioned, not to take the 




k- 



-15D 




Q 



4 



0>|o 



Fio. 360. 



0)|b 



•*W 



by the use of the wrench must be taken into consid- 
eration. Small bolts frequently break because of 
this strain. 

It is also necessary to take into account the man- 
ner in which the load is applied. As the nature 
of metal of the bolt may not be known as to its 



safe stress higher than 5,000 lbs. per sq. in. as given 
in the table. 

Fig. 360 shows the generally adopted proportions 
of a wrench. The wrench may be drawn for any 
size of a bolt head or nut, with the proportions of 
the parts as given in this illustration. 



ROGERS' DRAWING AND DESIGN. 



243 



RIVETS AND RIVETED JOINTS. 

For fastening together two or more comparatively 
thin pieces of metal, rivets are generally employed ; 
their greatest application is found in boiler work, 
where the joining of plates by riveting is found to 
be the only practical method. 

This method of fastening, however, is compara- 
tively expensive and unsatisfactory in many ways ; 
the rivets form a permanent fastening and can only 
be removed by cutting off one of the heads ; this 
creates trouble and expense. 

The process of punching the holes in the plates 
for riveting also has a serious effect by reducing the 
tensile strength of the plates by the disturbing in- 
fluence of the punch on the metal near the riveted 
joints ; for better work the holes are now generally 
made by drilling ; this, again, is more expensive, 
especially without the use of multiple drilling ma- 
chines. 

The injury due to punching, when the plates have 
not been cracked by the process, may be remedied 
by annealing them after punching; the ill effect of 
punching may also be removed by punching the 
holes ys" smaller in diameter than the required size 
of the hole, which may then be completed by ream- 
ing. 



Other injurious effects of punching are, i, the 
difficulty of correct spacing by this method, and 2, 
the fact that a punched hole is always tapered, the 
wider end of the hole being tha't next to the die. 




Fig. 361. 



Rivets are made in different forms ; that most 
commonly employed being of a spherical or cup head 
form, as illustrated in Fig. 361 ; both parts of this 
rivet show the spherical head. 



244 



ROGERS' DRAWING AND DESIGN. 



The rivet shown in Fig. 362 has a conical head, 
the lower part showing a pan head. The right pro- 
portions of the parts of the above rivets are given 
in the illustrations. 




Fig. 382. 



Fig. 363 shows a rivet with countersunk heads. 
The usual proportions of this kind of rivet are 



marked on the figure. 



In all the above illustrations the diameter of the 
rivet is taken as the unit of all proportions. 

The construction of the spherical head, Fig. 361, 
is as follows : 




Fio. 36.3. 



With a radius equal to one-half the diameter of 
rivet, from the center A on the vertical center line, 
describe a circle cutting the center line at the points 
B and C. Set the compasses to the distance BC 
and from the point B as center, describe an arc cut- 
ting the outline of the upper plate in the point D. 
Make BE equal to the distance AD and with E as 



ROGERS' DRAWING AND DESIGN. 



245 



center and CD as radius, describe the arc which 
forms the outhne of the spherical head. 

The construction of the other kinds of rivets may 
be easily understood from the illustrations without 
special explanation. 




The length of the rivet required to form the head 
is about i^ times the diameter of the rivet. For 
countersunk rivet heads, a trifle more than one-half 
of this amount is allowed. 

Riveted joints may give way because of the tear- 
ing of the plates between the rivets, as illustrated in 
Fig. 364, by breaking of the plates between the 




246 



ROGERS" DRAWING AND DESIGN. 




Fig. 3fi«. 




Fig. 3H7 



rivet holes and the edge of the plate, as shown in 
Fig. 365 ; by crushing of the plate or by crushing of 
the rivet, and by the breaking of the rivet through 
shearing, as indicated by Fig. 366. 

By the pitch of rivets is meant the distance be- 
tween the centers of two adjoining rivets, in a 
single riveted joint, that is where the seam is 
formed by one row of rivets, Fig. 367. When 
more than one row of rivets make the joint, the 
pitch is the distance between the center lines of 
rivets in the same row, Fig. 368. -; '" 



J=>itch 




Fig. 368. 



ROGERS' DRAWING AND DESIGN. 



247 




The distance between the centers of two adjoin- 
ing rivets, both in the same diagonal row is called 
the diagonal pitch, Fig. 369. 

The strength of a riveted joint depends upon 
the arrangement of the rivets and upon their pro- 
portions. 

Since a rivet may part either by shearing or by 
crushing, it is necessary for a given thickness of 
plate to find the proper diameter of a rivet having 









Fig. 370. 



T^i 



Fig. 3T1. 



f/ v^^/' ''^^'' 



248 



ROGERS' DRAWING AND DESIGN. 



equal shearing and crushing strength. The rela- 
tion between the thickness of the plate and the di- 
ameter of the rivet, calculated for single shear, is 










Fig. 373. 

expressed by the following formula;, of which the 
first is true for iron rivets and the second for steel 
rivets : 

d=2.o6 t for iron rivets. 

d=2.28 t for steel rivets. 





/A 



^'k 


^\ 


\^ 


.^ 



?^ 



^^ ft 



Fig. 374. 




Fig. 375. 



vS 



Fig. 376. 



ROGERS' DRAWING AND DESIGN. 



249 



d expresses the diameter of the rivet and t stands 
for the thickness of the plate. 

For plates thicker than f ^-in. the diameter of the 
rivet may be smaller in proportion to the thickness 
of the plate than is required by these formulae. 

The proportions commonly observed in practice 
for lap-joints and single-strap butt-joints is given in 
the following- table: 



Thickness of plate in inches. 
Diameter of rivet in inches 


A 1 i 

1 14 


i\ 1 f 


J_ 
i 


f 1 f 


7 

I 


I 


4 
H 



Numerous styles of riveted joints are in general 
use. The two classes into which the different styles 
may be divided are the lap-joint and the butt-joint. 

In the lap-joint, Fig. 370, the plates overlap each 
other. Figs. 371, 372 show other examples of this 
form of riveted joint. 

Fig. 373 shows a butt-joint. Here the plates are 
butted against each other and a cover plate or strap 
is placed over their junction and the rivets passed 
through the plates and strap. Fig. 374 shows a 
butt-joint with two cover plates. 

The examples of joints thus far illustrated differ 
as to the number of rows of rivets that are used for 
the seam. Fig. 370 is a single-riveted joint. The 
butt-joints shown in Figs. 373, 374, are also single- 
riveted. In a single-riveted joint the edge of each 
plate is pierced by only one row of rivets. 









-nSSr^ 



; 



Fig. 377. 




Fig. 378. 



I '^ 



; 



Si --U 



^1. 
I 

.1 _. 
I 



w- ^m-i-m^ 



Fig. 3.9. 



250 



ROGERS' DRAWING AND DESIGN. 




^--^ 



^ 




Fig. 380. 



Figs. 371 and 372 show double-riveted joints ; 
here the edge of each plate is pierced by a double 
row of rivets. When the rivets are opposite each 
other, as in Fig. 372, the seam is known as chain- 
riveted. 

When the positions of the rivets in one row are 
opposite the spaces between the rivets in the other 
row, the seam is staggered. 

The following illustrations are examples of riveted 
joints taken from practice in boiler work. 

Fig. 375 . A double-riveted lap-joint for two y^," 
plates, having -j-l" rivets, 1/% holes. The pitch of 
the rivets in this case is equal to 2^". 




Fig. 381. ,~^ 

A similar joint for S/i' plates is also shown in Fig. 
376. Here the pitch is 2^" and the rivets ir', the 
holes being made i ". Another joint of the same 

T 





FiG. 382. 



ROGERS' DRAWING AND DESIGN. 



251 




Fig. 383. 





/^Si'. 



im* 



Fici. 3M. 



Fig. 385. 

character is illustrated in Fig. 377. Here the 
plates and rivets are the same as in Fig. 376 ; 
the pitch, however, is 33^ ". 

In the double-riveted lap-joint shown in Fig. 
378, the plate is fl ', the rivets lyV'. the holes are 
made i|" and the pitch is 3". 

A butt-joint with double cover plates is illus- 
trated in Fig. 379. Here the plate is |" steel and 
i" rivets. The inner covering strap is J^g" thick, 
the outside strap is equal in thickness to that of 
the plate. 

Similar joints are shown in Figs. 380, 381, 
382, 383 and 384. 

The joint shown in Fig. 385 is not used very 
frequently. 



POWER TRANSMISSION. 



The oft-repeated word transmission comes from two Latin words, trans, across, or over, and mittere, 
to send, hence, to carry from one place to another; the illustration of a few devices for the transmission 
of power from its cause to its place of useful employment is the limit of this section of design. 

Prime movers or receivers of power, are those pieces or combination of pieces of mechanism which 
receive motion and force directly from some natural source of energy ; the mechanism belonging to the 
prime mover may be held to include all pieces which regulate or assist in regulating the transmission of 
energy, from the source of energy or power. 

Throughout this preliminary sketch, power and energy are used synonymously. 

The useful zvor k of the prime mover is the energy exerted by it upon that piece which it directly 
drives; and the ratio which this bears to the energy exerted by the source of energy is the efficiency of 
the prime mover; in all prime movers the loss of energy may be distinguished into two parts, i, 7ieces- 
sary loss ; 2, zvaste. 

The sources of power in practical use may be classed as follows : (a) Strength of men and ani- 
mals, (b) Weight of liquids, (c) Motion of fluids, (d) Heat, (e) Electricity and magnetism. The duty. 
of a prime mover is its useful work in some given unit of time, as a second, a minute, an hour, a day. 

Among the first examples of power transmission may be mentioned the case of a man hauling up 
weight with a rope or pushing or pulling an oar or capstan; in these instances the man is the prime 
mover and the duty performed is the raising of the weight and the moving of the vessel. 

The various combinations of mechanical 'povfers produce no force : they only apply it. They form 
the communication between the moving power and the body moved ; and while the power itself may be in- 
capable of acting except in one direction, we are able by means of cranks, levers, and gears, to direct and 
modify that force to suit our convenience. Every one may see examples of this in the construction of 
the most common pieces of machinery as well as in the most complicated. 

355 



256 



ROGERS' DRAWING AND DESIGN. 



SHAFTS. 

When a shaft is rotated by a lever attached to it, 
as in Fig. 3S6, or by a pulley or a gear-wheel as in 
Fig. 387, and a force P is applied to the free end 
of the lever or to a point at the rim of the pulley or 

u n ?; 




at the pitch-circle of the gear-wheel a twisting strain 
is produced on the shaft, this twisting strain causes 
a combination of stresses within the fibres of the 




Fig. 38: 



shaft, which mainly consist of shearing stress. The 
shearing stress is equal to nothing at the center of the 
shaft and it is greatest at its circumference. The 



twisting strain is obtained by multiplying the length 
of the lever, or the perpendicular distance from the 
point at which the force is applied to the center of 
the shaft, by the force P. If this distance be equal 
to R, Fig. 386, then R X P = T, which is called the 
twisting moment and is expressed in inch pounds. 

It is evident that the twisting moment must be 
equal to the resisting moment of the shaft. 

For finding the diameter of a crank shaft of a 
stationary engine with cylinders up to 30" in diam- 
eter some authorities recommend the following 
practical rule : 

The diameter of the crank shaft is equal to the 
radius of the cylinder minus 5^ of an inch. 

In practice many different diameters are found 
performing the same work. 

Now let T = twisting moment on shaft in inch 
pounds. N = number of revolutions of the shaft 
per minute. H = horse power transmitted, then the 
horse power equals 

2 X 3.1416 X T X N 



H 



: 0.00001587 



OJ'"- 



The number 33,000 in this formula expresses 
33,000 foot pounds of work performed per minute, 
and this amount of work is called one horse power. 

The above formula gives a method of finding the 
horse power transmitted by a shaft. 



ROGERS' DRAWING AND DESIGN. 



257 



Rule : Multiply the twisting moment in inch 
pounds by the number of revolutions per minute, 
and multiply the product by the number o.oooo- 
1587 the product will be the horse power trans- 
mitted by the shaft. 

Example : 

Find the horse power transmitted by a shaft mak- 
ing 100 revolutions per minute, provided with a 
gear wheel 36 inches in diameter (pitch circle), the 
turning force being 4,000 pounds. 

Solution : 

Multiply the pitch radius of the wheel,=i8 
inches by the force applied, ^4,000 pounds, and 
multiply the product by the number of revolutions 
and by the number 0.00001587: Horse power= 
18 X 4000 X 100 X 0.00001587=114.264. 

From the above formula the following expres- 
sions are obtained : 

12 X 33,000 X H 63025.21 X H 



2 X 3.1416 X N 



N 



From the same formula the twisting moment may 
be determined when the horse power transmitted by 
the shaft and the number of its revolutions are 
given. 



The number of revolutions may also be obtained 
from the same formula ; thus, 

12 X 33,000 X H 63025.21 X H 
^ "2 X 3. 14 1 6 XT~ '^ T 

Example : 

To find the number of revolutions which a shaft 
must make per minute in order to transmit 114.264 
horse power, when a force of 4,000 pounds acting on 
the pitch circle of a gear-wheel of 36" in diameter 
produces the twisting moment. 

Solution : 

The twisting moment in this case is equal to 
18 X 4,000= 72,000-inch pounds To find the num- 
ber of revolutions required divide the given horse 
power 114.264 by 72,000 and multiply the product 
by the number 63025.21 thus obtaining a quotient 
of 100.2 or the revolutions per minute. 

When the twisting moment only is to be consid- 
ered in calculating the diameter of a round shaft, 
which is to transmit a given horse power at a given 
speed, the following formula may be used : 

The cube of the diameter of the shaft or. 



D 



Twisting moment 



0.196 X stress in pounds per square inch. 



258 



ROGERS' DRAWING AND DESIGN. 



The stress is taken in pounds per square inch at 
the outer fibres of the diameter of the shaft. For 
steel shafts the stress may betaken at 10,000 pounds 
and for wrought iron at 8,000 pounds per square 
inch. 

Long shafts are subjected to combined twisting 
and bending actions. 

Let B = bending moment ; 
T == twisting moment ; 

Ti = the equivalent twisting moment. 

Then Ti = B + V^M^T^ 

In practice for long shafts in factories the follow- 
ing simple formula is recommended : 

D^ = 125 X horse power 



number of revolutions of shaft. 



The speed of the shaft depends upon the speed of 
the driving belt or by the diameters of the pulleys 
upon it. Shafts in machine shops are run from 
about 120 to 150 revolutions per minute; wood 
working machinery shafts usually run from about 
200 to 250 revolutions per minute. Shafts in woolen 
mills run up to 400 revolutions per minute. Line 
shafts should, as a rule, not be less than i^" thick 
in diameter. 

The distance between the centers of the bearings 
should not be great enough to permit a deflection 
of more than ^w" per foot of length. The more 



pulleys are on the shaft the closer the bearings must 
be. The beams may be placed about 8 feet apart, 
and each beam to be provided with a hanger on its 
lower side. To prevent end motion on shafts a 
collar is placed on each side of one of the bearings. 



JOURNALS. 

That part of a horizontal shaft which rotates in a 
bearing is called z. journal. The pressure of a shaft 
on a journal acts in a direction perpendicular to its 
axis. When the shaft is placed in an inclined posi- 
tion the pressure acts in a direction inclined to the 
axis of the shaft. The pressure of a shaft placed in 
a vertical position acts in the direction of its axis. 
The journal of a vertical shaft is called a pivot. 

The diameter of a journal must be made as small 
as the required strength will permit and as long as 
is necessary to keep the pressure per square inch as 
small as possible. This pressure per square inch is 
not measured on a circumference of the journal but 
by the area of its projection. 

Example: A journal 3" in diameter and 6" long 
will have a projected area of 3 X 6=18 square 
inches. Now if the pressure of the journal is 300 
pounds per square inch then the total pressure is 
equal to 18 X 300=5,400 pounds. 



ROGERS' DRAWING AND DESIGN. 



259 



Example : If the total pressure of a 3 " diameter 
journal equals 5,400 pounds and it is desired not to 
exceed a pressure of 300 pounds per square inch 
then the length of the journal is found thus : 

5,400 , . , 
^L-L = 6 inches. 

300 X 3 

Example : If a given journal is 3 inches in diam- 
eter and 6 inches long and its total pressure is 
known to be 5,400 pounds, then the pressure per 
square inch of projected area is found as follows: 
5,400 



X 6 



300 pounds. 



To find the pressure per square inch of projected 
area for a pivot bearing, multiply the square of the 
diameter of the shaft by .7854 and divide the total 
pressure of the shaft by the product thus found. 

The magnitude of pressure per square inch varies 
greatly in different cases in practice. It is gener- 
ally reduced where a greater speed is required. 
The maximum intensity of pressure on the main 
journal bearings of steam engines is 600 pounds per 
square inch for slow running and 400 pounds for 
high speed engines. Wherever possible it is ad- 
vantageous to make long bearings, thus reducing 
the pressure by about 200 to 300 pounds per square 
inch. 



Some manufacturers allow a pressure of 150 
pounds per square inch for cast iron journals for 
factory shafts. 

For pivot bearings the following pressures per 
square inch are given by a high authority, as being 
the most desirable. 

1. Wrought iron pi^-ot on gun metal bearing, 700 
pounds. 

2. Cast iron pivot on gun metal bearing, 470 
pounds. 

3. Wrought iron bearing on lignumvitae bearing, 
1,400 pounds. 

According to the latest practice it seems, how- 
ever, that, for pivots which have to run continuously, 
the above-mentioned pressures should be reduced 
to one-half. 



BEARINGS. 



The simplest form of a journal bearing for a shaft 
or spindle of a machine is simply a hole in the frame 
supporting the rotating piece. If it is necessary to 
increase the length of the bearing the frame must 
be made thicker in this particular place by casting 
bosses on it as shown in Figs. 388 and 389. Fig. 389 
is an end view and 388 is a section of such a bear- 
ing. The above described form of bearing is not 



260 



ROGERS' DRAWING AND DESIGN. 



durable as it has no means of adjustment for taking 
up the wear, and it cannot be renewed without re- 
newing part of the frame of the machine. It is 
therefore better to use the form of solid bearing 
shown in Figs. 390 and 391. In this case the hole 
is bored much larger than the journal, and lined 
with a solid bushing of soft metal, which can easily 
be replaced when worn. This arrangement requires 
a screw or key to hold the bushing in place ; in most 
cases the bushing is driven into the hole with con- 
siderable force to prevent it from turning, see Figs. 
390 and 391. 

Bearings for horizontal shafts have different 
names, which indicate the manner in which they are 
used. 






Fig. 390. 



Fig. 391. 




Fig. 38«. 



HANGERS. 

When a bearing, is suspended from the ceil- 
ing it is called a hanger. Figs. 392 to 395 
show the various details of a hanger made by 
a leading manufacturer. Fig. 392 is a side 
view ; Fig. 394 the longitudinal section. 
This design was first introduced by Sellers, 
and has been reproduced and modified by 
different manufacturers. It has a bearing 
box. Fig. 393, with a spherical center which 
is held between the ends of two hollow stems, 
all these parts are made of cast iron. 



Fig. 3m. 



ROGERS' DRAWING AND DESIGN. 



261 




Fio. 395. 



Fig. 393. 




262 



ROGERS' DRAWING AND DESIGN. 



These stems, Fig, 395, are provided with 
screw threads at their outer ends, ordinarily 
shallow square thread. The bosses on the 
frame are also provided with a similar screw 
thread, into which fits the screw of the 
stem. 

By means of the thread on the stems the 
height of the bearing can be adjusted, and 
the spherical centers allow a considerable 
adjustment in other directions. 

This construction makes the setting up or 
lining up of shafting much easier and the 
hangers made as described above enjoy 
therefore the greatest popularity at the 
present time. 



-^■- 



WALL BRACKETS. 

When a shaft is to be supported by a 
bearing fixed to a wall or pillar, a wall 
bracket is generally used for this purpose. 
In Figures 396 to 398 is shown a form of 
a wall bracket of an elegant and most solid 
design. 



nzj 



-aj 




..,._51'_ 



Figs. 393, 397 and ; 



ROGERS' DRAWING AND DESIGN. 



263 



PEDESTALS AND PILLOW-BLOCKS. 

The words pedestal, pillow-block, bearing and 
journal box are used indiscriminately. 

A bearing designed to support a shaft above a 
floor or any fixed surface is called a pedestal or 
pillow-block, depending upon the 
type of bearing, as will be seen 
later. A simple pillow-block is 
shown in Figs. 399 and 400. It 
consists of two parts, the box 
which supports the journal and 
the cap which is screwed down to 
the box by two screws or bolts, 
called cap-screws. 

Fig. 399 shows 
the front view of 
a complete pil- 
low-block with 
cap and bolts. 
Fig. 400 is a top 
view of base with 
the cap removed. 
The seats in the 
journal-box are 
usually babbit- 
ed, that is, lined 

Figs. ; 




with babbit, a soft metal whose composition is as fol- 
lows : One pound of copper, ten pounds of tin and 
one pound of antimony. To hold the babbit in 
place recesses are cast in the cap and base, extending 
almost across the entire width of the bearino-. The 
hangers as well as the wall brackets shown above 
have the bearings babbited in 
the same manner. The babbit 
is cast in as follows : A mandrel, 
having a diameter a trifle less 
than that of the journal, is placed 
in position within the bearing 
box which the shaft is to occupy. 
The babbit, in a molten state, is 
then poured around it and the 
bearing is then 
bored to the 
proper diameter. 

Instead of 
using babbit for 
the friction sur- 
faces in bearings 
other metals, 
such as brass, 
gun metal or oth- 
er alloys may be 
used. 



AND 400. 



264 



ROGERS' DRAWING AND DESIGN. 



The melting point of these metals, however, is so 
high, that they cannot be poured into the box and 
cap directly, as in the case of a babbited bearing. 

They consequently are made as separate pieces, 
called steps or brasses, and are fitted into the box 
and cap in different ways. 

In some cases, where 
very little, and slow 
motion is required, the 
method described in 
Figs. 390 and 391 may 
be employed. 

The bearing in this 
case is made by boring 
a hole through the cast- 
ing, and the brasses 
consist of a simple 
sleeve, which is called 
a bushing. 

This bushing is sim- 
ply turned off and 
bored, and is then 
forced into the hole. Bearings that are made in 
halves, for the taking up of wear, and for removing 
the shafts, are fitted with two brasses however, the 
bearing and the cap brass. In this case the outside 
of the brasses is often made square or octagonal, 




fitting into recesses of similar shape in the box and 
cap of the pillow-block. This is done to prevent 
them from turning with the .shaft. 

To prevent the brasses from sliding out endwise, 
they are provided with a shoulder on each end, 
which fits over the ends of the bearing-. Brasses of 

the octagonal type as 
well as their applica- 
tion are fully illustrated 
in Figs. 402 to 404. 
When a shaft is to~be 
supported a considera- 
ble distance above the 
floor, the pillow-block 
is placed on a stand- 
ard. 

This standard may 
be cast separate from 
the pillow-block, and 
in this case both are 
fastened together by 
bolts. 



Note. — Brasses are made from different alloys which vary accord- 
ing to the judgment of the designer. Some engineers recommend the 
following composition : Six pounds of copper, one pound of tin and to 
every hundred pounds of this mixture one-half pound of zinc and one- 
half pound of lead are added. 



ROGERS DRAWING AND DESIGN. 



265 






[f^-'lr--^ '%' *--^---H 



V^ 




266 



ROGERS' DRAWING AND DESIGN. 



Very often, however, the pillow-block and stan- 
dard are cast in one piece and it is then called a 
pedestal. 

A pedestal is shown in Figs. 402 to 404. Fig. 
403 is a front elevation and 402 shows its top view. 
Fig. 404 represents a side view of the pedestal. 

The various parts of this pedestal are exactly the 
same as in the above-described pillow-block with 
the exception that the seats or steps in this case are 
of an octagonal shape. 

There is no established standard of proportions 
for the parts of a bearing ; the proportions of pillow- 
blocks made by different manufacturers vary con- 
siderably. 

A pedestal for supporting a very small shaft is 
often obtained by turning a hanger upside down, 
reversing, of course, the bearing. 

Such small pedestals are usually called Jloor 
stands. 

The main bearings of large engines with girder 
beds are also often called pedestals. 



BELTS AND PULLEYS. 

Belts most commonly used are made of leather ; 
they may be single or double ; in damp places, 
canvas belts, covered with rubber are sometimes 
used ; leather belts are usually run with the hair 
side on the outside or away from the pulley. 
Long belts when running in any other than a verti- 
cal direction, will work better than short belts, as 
their own weight holds them firmly to their work. 

Fig. 405 shows an open belt, and Fig. 406 a crossr 
belt. 

Pulleys connected by open belts run in the same 
direction, while those connected by cross-belts run 
in opposite directions. When two pulleys are con- 
nected by a belt, the motion of one, the driving 
pulley, is transmitted to the other pulley, the follower. 
If we assume that there is no stretching or slipping 
of the belt, every part of the circumference of the 
follower will have the same velocity as the driving 
pulley being equal to the velocity of the belt passing 
over them. 

If the pulleys are of different diameters, for in- 
stance, if the driver has a diameter two times greater 
than the diameter of the follower, the latter will 
make two complete revolutions for each revolution 
of the driver. This ratio between the speeds of two 
pulleys is expressed in the following 



ROGERS' DRAWING AND DESIGN. 



267 




Fig. 405. 





Fio. 407 



Fig. 406. 



268 



ROGERS' DRAWING AND DESIGN. 



Rule : 

The tiumber of revolutions of two connected pulleys 
are inversely proportional to their diameters. 

This may be expressed in the following formula : 

Number of revol. diameter of second pulley 
of nrst pulley 

Number of revol. Y)\2.m^^^v of first pulley 
of second pulley 

Example : 

A pulley 40" in diameter, making 300 revolutions 
per minute, drives a second pulley 20" in diameter. 
How many revolutions per minute does the second 

pulley make ? 

40X300 

No. of revol. of second pulley == ^600 revol. 

^ ^ 20 

To find the revolutions of the follower : 
Multiply the diameter of driver by its number of 

revolutions, and divide the product by the diameter of 

the follower. 

Example : 

The follower is 20" in diameter and makes 1 50 rev- 
olutions. What is the size of the driver used on a 
driving shaft that makes 200 revolutions per minute ? 

Diameter of driver =-2 = i5 inches, that is, 

200 



the diameter of the driver is found by multiplying 
the diameter of the follower by its number of revo- 
lutions, and dividing the product by the number of 
revolutions of the driver. 

For four pulleys connected by belts, as shown in 
Fig. 407, the following rule is to be applied : 

The number of revohitions of the first pulUy, tnul- 
tiplied by the diameter of each of the driver.-, equals 
the num.ber of revolutions of the last pulley, miilti- 
plied by the diameter of each follower. 

Example : 

Let the diameter of the drivers be 40" and 30", 
the diameter of the first follower 10" and of the 
second follower 15". What is the number of revo- 
lutions of the last shaft, when the first shaft makes 
100 revolutions per minute ? 

Here the speed of the last shaft, multiplied by the 
diameter of the followers, 10" and 15', must equal 
the speed of the first shaft, 100 multiplied by the 
diameter of the drivers, 40" and 30" ; that is, 

speed of last shaft X 10 X 15 = 100 X 40 X 30, or 

1 f 1 1 r 100 X 40 X ^o „ , 

speed 01 last snait = ^^ = 800 revol. 

^ 10 X 15 

When the number of revolutions of the first and 
last shafts are known, and it is required to find the 
diameters of the pulleys, apply the following 



ROGERS' DRAWING AND DESIGN. 



269 



RULH : 

Divide the higher number of revolutions by the 
lozver. 

In a case where four pulleys are to be used, 
we find the numbers whose product is equal to 
the quotient resulting from the above division of 
the speeds. One of these numbers is taken as the 
ratio of the diameters of one pair of the pulleys, and 
the other number, of the other pair. 

Example : 

It is required to run the last shaft with a speed of 
1,500 revolutions, the driving shaft making 300 
revolutions per minute. What size of pulleys are 
required when four pulleys are to be used ? 

The quotient resulting from division of the two 

speeds, equals — ^ — -^ 5. Two numbers whose 

product equals 5 are 2)^ and 2. Consequently one 
pair of the pulleys must betaken in the ratio of 2^4 
to I and the other pair as 2 to i. Therefore, the 
first pair may be 30" and 12" and the other pair 24" 
and 1 2 " 

To find the speed of the belt : 
Multiply the circumference of the pulley by the 
number of revolutions per minute. 



Example : 

Let the diameter of the pulley be equal to 2 ft. 
and the number of revolutions per minute 100. 
Then, 2 X 3.14 X 100 -= 628 ft. per minute, the 
speed of the belt. 

The relation between the speed of the belt in feet 
per minute, the width of the belt in inches, and the 
horse power to be transmitted, is expressed in the 
following practical formulae : 

The horse power to be transmitted is found, by 
multiplying the speed by the width of belt and divid- 
ing the above product by goo ; or 

To find the required width of the belt, multiply 
the horse power to be transmitted, by goo, and divide 
the product by the speed ; or 

To fijtd the speed in feet per minute, multiply the 
horse pozoer by goo, and divide the product by the 
width of the belt. 
Example : 

Two pulleys, each 2 ft. in diameter, connected by 
a belt, make 200 revolutions per minute. It is de- 
sired to transmit 20 H. P. What is the proper 
width of the belt to be used ? 

The speed of the belt is equal to 2 x 3.14 x 200 
= 1,256 ft. per minute; consequently the width of 
the belt equals 

2 = 14.6 inches, or a belt 14^ inches. 

1.256 



270 



ROGERS" DRAWING AND DESIGN. 



The above formulae are true of a single belt. 
When double belts are used, made of two single 
belts cemented and riveted together through their 
entire length, they should be able to transmit twice 
as much power as a single belt, and even more. 

The above formulae may be applied to the calcu- 
lation of double belts, provided the number 630 is 
put in the formula instead of the constant number 
900. This will give the required proportions for 
belts, when used upon small pulleys, in which case 
more power is required for the transmission. 



SPEED OF MACHINE TOOLS. 

I n selecting the speed of pulleys, the designer must 
be guided by the speed of the machine which is to 
be driven. 

The speed of different machines varies according 
to the work which they perform, as, for example, the 
cutting speed of machine tools, or the velocity of 
emery wheels. 

Grindstones in machine shops, suitable for grind- 
ing machinists' tools may be run with a peripheral 
speed of about 900 ft. per minute ; grindstones for 
pattern makers' use, run about 600 ft. per minute. 

Emery wheels may be run with a peripheral veloc- 
ity of 5,500 ft. per minute. 



Polishing wheels, such as leather-covered wooden 
wheels, or rag wheels, may run with a peripheral 
velocity of 7,000 ft. per minute. 

The speed of cut for cast iron is 20 to 30 ft. per 
minute, for tool steel about 10 ft. per minute. Cut- 
ters in the milling machine may be run with a per- 
ipheral velocity of 80 ft. per minute for gun metal ; 
35 to 40 ft. for cast iron ; and for machine steel 
about 30 ft. per minute. 

Example : 

What is the proper number of revolutions of tKe 
spindle of a machine shop grindstone 24" in diameter? 
The usual peripheral speed is 900 ft. per minute. 
The circumference of the given stone is equal to 
2 ft. X 3.14=6.28 ft. 

^ — r=about 143 revolutions per minute 
6.28 ^■^ ^ 

Example : 

Let the emery wheel in a grinder be 12" in diam- 
eter, and let it be required to run the wheel with a 
peripheral velocity of 3,600 ft. per minute. What 
should be the speed of the spindle of this bench 
grinder ? 

The circumference of the wheel is 3. 14 ft. Divide 

3,600 by 3.14 and the speed of the spindle is found. 

3,600 



3-14 



= about 1,150 revol. 



ROGERS' DRAWING AND DESIGN. 



271 



The following are rules recommended by practical 
experience for the use of belts. 

Pulleys of small diameter, say of less than i8", 
should not be used for double belts. Narrow, thick 
belts work better than thin ones. If wide belts are 
used, it is proper to increase their thickness. 

This, however, is only true within certain limits ; 
the tendency among engineers is to go to the ex- 
treme in this direction ; it depends largely upon the 
class of work the belt is to be used for, and the 
only wa}' anyone can claim to be expert in this line 
is through practical experience and good judgment. 

The weakest part of the belt is at the joint ; for 
this reason joints should be made very carefully 
according to the most approved methods ; the same 
fastening does not answer for all belt-joinings. 

It is not advantageous to place two pulleys con- 
nected by a belt too near one to the other. A distance 
of 15 ft. between the shafts for narrow belts running 
over small pulleys is a good average. Wider belts 
running over larger pulleys for good work require 
a greater distance between the shafts. 30 ft. is a 
good average for such cases. The distance between 
the shafts should not be made too great, as this may 
cause too much of a sag of the belt, which may pro- 
duce such a pressure on the journals of the shaft 
as to injure them. 



Running belts in a vertical direction should be 
avoided whenever possible. Machine tools driven 
by vertical belts require particularly good well- 
stretched leather belts, which must be kept very 
tight. 

In tightening belts it must be remembered, that 
while tightening the belt, the pressure on the bear- 
ing is also increased, causing greater friction and 
wear on the bearing, especially with overhung 
pulleys. 

The angle of the belt with a horizontal line should 
not exceed 45° whenever possible. Belts are not 
run advantageously when their speed exceeds 2,500 
ft. per minute. 



PULLEYS. 



The rim of a belt pulley may be made either 
straight, Fig. 408, or convex, as in Fig. 409. It 
would seem that the belt would remain on the 
straight pulley more readily than on the convex one. 
Experience shows, however, that the belt always 
tends to run on the highest part of the pulley, pro- 
vided it does not slip, in which case the belt will fall 
off more readily from a convex surface than from a 
straight rim of the pulley. 



272 



ROGERS' DRAWING AND DESIGN. 



^^^\\^^ 



kmkkkk^ 




Fig. 408. 



Fio. 4(S. 




Fig. 410. 



Fig. 411. 



The flat or straight rim pulley is used where it is 
necessary to move the belt from one side of the rim 
to the other, as in the case where one pulley drives 
a pair of fast and loose pulleys. 

Whenever there is frequent slipping off" the rim 
of a belt, through a temporary increase in resist- 
ance, the pulfey is provided with flanges, as shown 
in Fig. 410. 

The amount of curvature in a section of the rim, 
is made greater, the faster the speed at which it runs. 
The curve may be an arc described with a radius' 
equal to about . 5 times the breadth of the pulley. 
The breadth of the pulley is generally made a little 
wider than the width of the belt, Fig. 411. The 
thickness of the rim at the edge may be found by 
dividing the diameter of the pulley by 200 and add- 
ing ys of an inch. For a pulley 25" in diameter, the 
thickness of the rim should be 
25 inches 



V^ inch == i^ inch. 



200 



The thickness of the walls in the central part of 
the pulley, called the hub. is found by a formula 
given by Mr. Thomas Box, as follows : 

Thickness of hub = 1 1- ^, where D is the 

96 8 

diameter of the pulley and d the diameter of the 

shaft. 



ROGERS" DRAWING AND DESIGN. 



273 



Prof. Unwin gives the following formuls;: 
For a single belt, the thickness of hub = 0.14 
V"B"D~ + 14 in. 

For a double belt, the thickness of hub = 0.18 



V B D -(- J4! in. where B indicates the breadth of 
the pulley. The length of the hub is made from 
73 B up to B. This is true for fast pulleys only. 

The hubs in loose pulleys are usually longer than 
in fast pulleys. The hubs in loose pulleys need not 
be so thick, and they project about ^ inch beyond 
each side of the face of the pulley. Fig. 412 shows 
loose and fast pulleys 




?^W^??^l////M; 



22Z^ 



Fig. 412. 




Fig. 413. 




Fig. 414. 



274 



ROGERS' DRAWING AND DESIGN. 




,m,»m;^'^7777m 



Fig. 415. 



Fig. 416. 



The arms of pulleys are usually straight, but 
sometimes they are curved, as shown in Figs. 413 




Fig. 417, 



and 414. It is the general practice in machine 
shops to draw the section of a pulley, as shown in 
Fig. 416, no matter what shape the arms may have. 



The straight-armed pulley is simplest in appear- 
ance and construction. There is no fixed rule for 
the number of arms in a pulley. Usually those up 
to 18" in diameter have four arms, and those of 
larger diameters, si.x arms. 

The cross-section of the arms of cast-iron pulleys 
is generally oval-shaped and of the proportions 
shown in Fig. 417. 

The longest a.xis of the oval a, may be found 
from the following practical formula;: the breadth_a 
being taken at the center of the pulley, supposing 
the arm to be continued through the hub to that 
point. B D 



4N 
B D 

2 N 



for sintrle belts, and 



for double belts. 



In these formulae B is the width of the pulley, D 
is its diameter and N the number of revolutions per 
minute. 

The proportions of the section of the arm near 
the rim may be two-thirds of the proportions given 
in the above formulae. 

It will be noticed that the breadth of the oval is 
given in the cubic power. To find the actual 
breadth a, multiply B by D, divide the product by 
4 N and then find the cube root of the resulting 
number. 



ROGERS' DRAWING AND DESIGN. 



275 



For varying the velocity of a shaft, speed cones 
are used, Fig. 418. As the belt will have a tendency 
to climb a conical pulley, special provision must be 
made for keeping the belt in place. It is also desir- 
able to have both cones alike, so that they can be 
cast from one pattern. 

Cone pulleys or speed pulleys, are frequently 
made in a series of steps, as shown in Fig. 419, in 
which case they are termed step-pulleys. 

It is an established fact, that when two cones are 
placed with their centers at a given distance, and are 
so related that the sum of their radii remains con- 
stant, an endless cross-belt, containingr both cones 
will not change in length in the smallest deeree 
during the change in the actual diameter of each 
cone. 

It is necessary to keep in mind the fact that the 
sum of the radii of both cones and the distance be- 
tween their centers remain constant. As a result 
of this the sum of the radii of two opposite pulleys 
in a series of steps must be the same for all steps, 
as only with this condition will a crossed belt fit any 
pair of pulleys in the series. 

Fig. 420 shows three sets of pulleys which may 
be arranged into a step pulley with three sets or 
steps. The distances between each set of pulleys is 
the same, and the sum of the diameters of the 



L*__ 



pulleys in each one of the three sets is also ihe 
same. As a result of these conditions the length of 
the crossed belt for all sets is the same. 




Fig. 418. 



The above statement does not hold true for open 
belts. The middle sections of cone pulleys for open 



276 



ROGERS' DRAWING AND DESIGN. 



belts must be larger proportionately than for crossed 
belts. 

In the pair of cone pulleys shown in Figs. 421 and 
422, both are made alike, and the first one makes N 
revolutions per minute ; let it be required that the 



equal the small diameter multiplied by the square 
root of the quotient of m divided by n. 

As was remarked before, for open belts, the middle 
diameter of the cone pulleys must be made larger. 
If D and d are the large and small diameters of a 




Fio. 419. 



second pulley should have a range of speed from 
m to n revolutions, m beine the greater number. 
Then N must equal the square root of the product 
of m and n, thus, N ^ Vm x n. 

The large diameter in the cone pulleys must 



cone pulley, then the proper middle diameter is 

1 , D + d o'o8 (D-d)= , r ■ .u A- 
1 to -I- ^^?; -, where L is the dis- 



equal to 



+ 



2 ■ C 

tance between the two shafts. 

When the middle diameter is thus found, the 



ROGERS' DRAWING AND DESIGN. 



«vy 



outline of the cone is laid out by an arc of a circle 
passing through the ends of the diameters D and d 




as well as the ends oi the middle diameter. When 
it is desirable to substitute a step pulley for a con- 



tinuous cone, A B D C, Fig. 423, the cone is divided 
into the required number of equal parts by parallel 




Figs. 421 and 422. 

lines, like E F, etc., drawn at equal distances. 
These diameters are then taken as center lines for 
the different steps. 



278 



ROGERS' DRAWING AND DESIGN. 



GEAR WHEELS. 

When two wheels with parallel axes, as shown In 
Fig. 424, are placed firmly together so as to form a 
rolling contact, the motion of one wheel, if there is 
no slipping, will produce a motion in the other 





Fig. 423. 



wheel ; in this case a point on the rim of one wheel 
will travel exactly at the same rate of speed as any 
point on the rim of the other wheel ; a rotation of 
this kind is called a positive rotation. Both wheels 
when in positive rotation by rolling contact, will 



have the same ratio of velocity, or as it is generally 
termed a constant velocity ratio. 

The number of revohitions of the shafts will be 
inversely proportional to the diameters of the wheels, 
and this ratio will remain constant, provided there is 
no slipping. 

With wheels having smooth surfaces it is im.pos- 
sible to maintain a constant velocity ratio, hence, to 




Fig. 424. 



secure this condition, the wheels are provided with 
teeth which will enable them to rotate without the 
possibility of slipping. 

To avoid a separate velocity for each tooth and 
to obtain an equal speed velocity in all parts of the 
wheel, the teeth are designed with proper proportions, 
which will be explained and illustrated hereafter. 



ROGERS' DRAWING AND DESIGN. 



279 






/ 




Fig. 425. 



r- 



280 



ROGERS' DRAWING AND DESIGN. 



The rims of two imaginary wheels which have the 
same axes and which would have the same velocity 
ratio as two given gear wheels and the same width, 
form what are z^\t.d, pitch surfaces ; the circles rep- 
resenting the section of both pitch surfaces, at right 
angles to the axes, are called the pitch circles. 

The part of a tooth in a gear wheel outside of 
the pitch circle is called the addendum, and the part 
of the outline or curve of the tooth on the addendum 
is called the face of the touth, as shown in Fig. 425. 

That part of the tooth inside of the pitch circle is 
called the dedendum, and the part of the surface of 
the tooth inside of the pitch circle forming the front or 
back of the dedendum is called the flank of the tooth. 
The point where the flank and the face meet is called 
the pitch point and is situated on the pitch circle. 
The circle passing through the tops of the teeth is 
called the addendum circle and is equal in diameter 
to the blank or disc, from which the gear is to be cut. 

The circle passing through the bottom of the teeth 
\'s,Z'A\&A\h.& dedendum circle. The distance measured 
on the pitch circle between the pitch points of two con- 
secutive teeth is called the circular pitch of the gear 
wheel. The circular pitch includes one thickness of 
tooth and one space between teeth ; the circular pitch 
is equal to the circumferetice of the pitch circle 
divided by the number of teeth in the gear wheel. 



Example : 

If the diameter of the pitch circle is equal to D. 
The circumference of the pitch circle is equal to 
3.1416 X D. Let the number of teeth in the wheel 



be N. Then 



3.1416 X D 

N 



is the circular pitch. 



^y diametral pitch is meant the number of teeth 
in the gear per one inch of its pitch circle diameter. 

Example : 

If the diameter of the pitch circle is equal to D 

Inches, and the number of teeth equals N ; then 

N 

iy= diametral pitch. 

The diametral pitch expresses in a direct and sim- 
ple manner the ratio between the diameter of the 
pitch circle and the number of teeth. Usually it may 
be expressed by a whole, number and therefore its 
form is convenient for expressing the proportions of 
the teeth, which are usually dependent upon the 
pitch, for this reason nearly^all gear calculations are 
made in terms of the diametral pitch. 

Rule : 

To change the diametral pitch to circular pitch, 
divide J. I ^16 by the diametral pitch. To change the 
circular to diametral pitch, divide j. 1^16 by circular 
pitch. 



ROGERS' DRAWING AND DESIGN. 



281 



The proportions commonly adopted for gears 
made with precision, are as follows : 

The addendum equals i divided by the diametral 
pitch. 

ExAMPLii: : 

If the diametral pitch equals 2 then the adden- 
dum equals yi. The pitch circle diameter, plus 
tivice the addendum equals the blank diameter of the 
S^ear. 

The dedendun is equal to the addendum, plus the 
bottom clearance. The clearance is generally equal 
to To the thickness of the tooth, measured on the 
pitch line. 

The thickness of the tooth and width of space, 
measured on the pitch circle, are each equal to one-half 
the pitch, on carefully cut gears ; in practice, how- 
ever, it is customary to make the width of the space 
slighth" larger than the thickness of the tooth, in 
order to allow for inaccuracies of workmanship and 
operating, unavoidable because of the difificulty of 
producing theoretically correct gears. This is par- 
ticularly necessary in cast gears. The difference 
between the thickness of the tooth and the width of 
space is called the back lash ; the amount of back 
lash necessary for a gear must be left to, the best 
judgment of the designer ; in cast iron gears it is 



\ 



sometimes equal to yV of the circular pitch ; this is 
good practice only for very rough castings. 

The flank and the dedendum circle are joined by 
small arcs, to avoid sharp corners at the root of the 
tooth. These are called filets and are usually made 
with a radius equal to one-seventh of the distance 
between two consecutive teeth, -measured on the adden- 
dum circle. 

When two gear wheels with parallel shafts are 
turning one the othelv^the distance between the 
centers of the shafts is eq\lal to the sum of the pitch 
diameters of both gears divided by 2. 

Example. — Let D equal the pitch diameter of 

one wheel and d the pitch diameter of the other 

wheel ; then the distance between the centers in this 

• f u 1 ■ D + d 
pair 01 i^ear wheels is — 

The number of teeth is found by dividing the cir- 
cumference of the pitch circle by the circular pitch. 
If the diametral pitch be given, the number of teeth 
is found by multiplying the pitch diameter by the 
diametral pitch. 

The pitch diameter is found by multiplying the 
num,ber of teeth by the circular pitch and dividing 
the product by j. 1^16. 



282 



ROGERS' DRAWING AND DESIGN. 



If the diametral pitch is given, the pitch diame- 
ter can be found by dividing the number of teeth by 
the diametral pitch. 

The diameter of the blank equals the pitch diame- 
ter plus 2 divided by the diametral pitch. 

If the number of teeth and the diametral pitch are 
known, add 2 to the number of teeth and divide by 
the diametral pitch. 

Gears may be classified as follows : 

Spur Gears for shafts which are parallel ; in this 
case the pitch surfaces would be cylinders. 

A gear wheel with a comparatively small number 
of teeth is termed a pinion. Bevel gears for con- 
necting shafts which intersect when lengthened. 
The pitch surfaces in this case are cones. Bevel 
gears of the same size connecting shafts at right 
angles are called miter gears. 

If the shafts are neither intersecting nor parallel, 
the pitch surfaces will be hyperboloids of revolution 
and the gears are called hyperbolic or skew gears. 

In all gears enumerated up to the present, the 
teeth are made with rectilinear elements and the 
pitch surfaces touch each other along straight lines. 

Worm Gears are used for connecting shafts 
which are at right angles to each other and which 
do not meet when lengthened indefinitely in either 



direction. The pitch surfaces meet in spiral lines. 
When the pitch circle is made with a diameter in- 
definitely increased, it will become a straight line 
and the gear is termed a rack. 

There are two kinds of teeth generally used and 
classified according to the methods of producing 
them ; these are involute teeth and cycloidal teeth. 

The cycloidal system of gearing was, for a long 
time, used almost exclusively ; of late, however, the 
involute system is rapidly gaining in popularity and 
many engineers advocate its general application in 
all cases. 

Involute teeth are of greater strength and will run 
well with their centers at varying distances and still 
transmit uniform velocity. The chief objection that 
has been raised against involute teeth is the obliquity 
of action, causing increased pressure upon the bear 
ings. 

If a flexible line be wound around a circle, and 
the part which is off the circle is kept stretched and 
straight, any point in it will describe a curve which 
is the involute of the circle. 

To draw the involute to a given circle, PABCO, 
Fig. 426. Divide the given circle into any number 
of equal parts by the points P, A, B, C, D, etc., 
through which points draw tangents to the circle. 
Make the length Aa equal to the length of one 



ROGERS' DRAWING AND DESIGN. 



283 



division AP, the length Bb equal to two such divi- 
sions Cc to three divisions, Dd to four and so on. 
Through the points P, a, b, c, d, etc., draw the 
required curve, which is a portion of the circle's 
involute. 

The outline of an involute gear tooth is 
made with a single curve, the involute of an 
especially selected circle which is called the 
base circle. The center of the base circle 
lies in the center of the pitch circle, and the 
base circle is always smaller than the pitch 
circle. 

Different manufacturers make the base 
circle of an involute gear of different diame- 
ters. Brown and Sharp make the diameter 
of the base circle equal to 0.968 of the pitch 
circle. The ordinary method of finding the 
base circle is as follows: 

If H is the center of the wheel and P the 
pitch circle, draw the addendum and deden- 
dum circles, Fig. 427. 

Take any point O, as the pitch point on 
the pitch circle and draw a radial line HH through 
this point. Draw a line EE making an angle of 
75° with the radial line HH. The base circle is 
found by drawing inside of the given wheel, a circle 
tangent to the inclined line, EE (which line is 



called the line of action). Let the base circle 
intersect the line H H at the point W. From this 
base circle the involute is drawn, passing through 
the point W and extending to the point V on the 




Fig. 426. 



addendum circle. The part of the flank between 
the base circle and the dedendum circle is straight 
and is part of the radius of the circle. 

No wheel having less than 12 teeth will gear cor- 
rectly together when the base circle is laid out in 



384: 



ROGERS' DRAWING AND DESIGN. 




Fw. 427. 



this manner ; in practice a curve wliich approximates 

the required involute curve is generally employed. 
In the Brown & Sharp system the line of action 

is drawn so as to make an angle of 75 3^°, Fig. 428. 

This is true for gears having more than thirty teeth : 
for gears having a smaller number of teeth, 
special rules are followed. 

The Brown & Sharp method explained 
above cannot be used for involute gears 
having less than thirty teeth, as the space 
left at the root is too narrow for the free 
motion of the mating gear. In such cases 
the curve is drawn from the base circle to 
the addendum ; from the base circle to the 
dedendum circle, the flank is drawn parallel 
to a radius of the wheel through the middle 
of the space between two adjoining teeth, 
being joined to the dedendum circle by ah arc or fillet. 

In an involute rack, in many shops, the teeth are made with 
straight lines passing through the pitch points of the teeth, as shown 
in Fig. 429. The direction of the straight edges of the teeth is at 
right angles to the line of action, that is generally lines making 
angles of 75° with the pitch line. For racks which are to run with 
pinions having fewer than thirty teeth, the outline of the teeth on 
the rack near the addendum are rounded to prevent interference with 
the flank of the pinion tooth. In the cycloidal system of gearing the 
outline of a tooth is made by a double curve ; here the face is a por- 



ROGERS' DRAWING AND DESIGN. 



285 




Fig. 428. 



tion of an epicycloid and the flank a hypocycloid, 
both joined in the pitch point. 

If a circle is made to roll along a straight line, 
always remaining in the same- plane, a point in the 
circumference of the rolling circle will describe a 
cycloidal curve. The rolling circle is called the gen- 
erating circle or describing circle. 



.cf/'/^v" 




Fig. 429. 



If the generating circle rolls along a straight line 
it will describe a cycloid. If the generating circle 
rolls along the outside of a circle it will describe an 
epicycloid, and when rolling along the inside of a 
circle, it will describe a hypoc\'cloid. 

The construction of these curves is shown in Figs. 
430, 431 and 432, 

To draw the cycloid in Fig. 430, draw a straight 
line AC. Describe the generating circle and divide 
it into any number of equal parts by the points i, 2, 
3, 4, etc. From B, the point of contact of the gen- 
erating circle with the straight line, set off distances 
equal to the portions of the circle, so that BC will be 
equal in length to one-half of the circumference of 
thfe rolling circle and will be divided into the same 
number of equal parts by the points i', 2',3', etc. 

Through these points draw lines perpendicular 
to the line ABC. Through the center of the 
generating circle draw a line parallel to the line 
AC ; this line will cut the perpendicular in the 
points a, b, c, d, e, etc. With these points as 
centers, describe arcs of circles with a diameter 
equal to the diameter of the generating circle. 
These arcs will touch the line AC in the points 
i', 2', 3', 4', etc. Through the point 2 on the 
generating circle, draw a line parallel to AC, cut- 
ting the arc which passes through i'. 



286 



ROGERS' DRAWING AND DESIGN. 



Through point 3 in the generating circle, draw a 
line parallel to AC and cutting the arc which passes 
through the point 2'. Through the points 4, 5, 6, 
etc., in the generating circle, draw parallels to meet 
the arcs cutting the points 3', 4', 5', etc., respectively. 
The intersections of these lines with the arcs deter- 
mine the required curve. 

Fig. 431. To draw the epicycloid, describe the 
generating circle tangent to the circumference of the 
given circle at the point B. Divide the generating 
circle into any number of equal parts by the points 
1, 2, 3, 4, etc. Set ofi the equal portions of the 
generating circle on the circumference of the given 
circle by the points i', 2', 3', etc., through which draw 



radial lines extended outside of the base circle AC 
and cutting them in the points a, b, c. d, etc., by a 
circle having one common center with the base 
circle and passing through the center of the gener- 
ating circle O. 

With the points a, b, c, d, etc., as centers, draw 
arcs with a radius equal to the radius of the gener- 
ating circle, these arcs touching the base circle in 
the points 1', 2', 3', etc. Through the points i, 2, 3, 
etc., in the generating circle draw arcs concentric 
with the base circle AC, to meet the arcs touching 
the points i', 2', 3', etc., respectively, and through 
these points of intersection draw the required 
epicycloid. 




Fig. 430. 



288 



ROGERS' DRAWING AND DESIGN. 



The hypocycloid is drawn in the same manner, as 
shown in Fig. 432. 

To lay out the outlines of a cycloidal gear, draw 
the pitch circle, Fig. 433, and divide it into a number 
of equal parts, corresponding to the number of teeth 
required. Each one of these parts is equal to the cir- 
cular pitch of the wheel. Bisect each one of these 
pitch distances to obtain the thickness of the tooth 
and the width of the space between the teeth. 
Wherever necessary make the space larger than the 
thickness, thus providing for back lash. Next, 
draw the addendum and dedendum circles, and select 
the proper describing circle. The profile of the 
tooth between the dedendum and the pitch circle, 
the face of the tooth, is made by an epicycloid gen- 
erated by the generating circle as it rolls along the 
circumference of the pitch circle. The flank, that 
is the outline of the tooth between the pitch circle 
and the dedendum circle, is a hypocycloid of a 
generating circle equal to the above generating circle, 
or, if convenient, with a generating circle having a 
different diameter. 

For two gears which are to run together, as in Fig. 
434, the faces of the teeth in both wheels must be 
described by one generating circle. Wherever one 
generating circle is used for the face as well as the 



flank of one wheel, the same generating circle should 
be used for both face and flank of the teeth in the 
mating gear. 

When the outline of one tooth is found, a template 
of thick paper may be cut to one of its sides and by 
attaching this template to an arm of suitable length, 
which may be held to the center of the wheel by a 
-pin, we can swing it around and bring it in position 
to draw the profiles of the rest of the teeth. 

Since it would be too much to describe all teeth 
by tracing for each one of them the proper cycloidal 
curves, it is usual to approximate these curves by 
means of circular arcs. We find an arc which very 
closely coincides with the proper curve for the face, 
and the same is done for the flank. The centers of 
these arcs are found by trying with the compass 
until the proper arc is found. 

In Fig. 435, the point A is found to be the center 
of an arc which very closely coincides with the flank 
of the tooth ab. Draw a circle through A, concen- 
tric with the pitch circle. The centers of all arcs 
for the flanks of the rest of the teeth in this gear 
will lie in the circumference of this circle, and the 
radii of these arcs will equal in length Aa. To draw 
the flank d on the tooth cd, set the compasses to a 
radius equal to Aa, put the needle point in d and 



ROGERS' DRAWING AND DESIGN. 



^.— . *^.~ 



289 




Fig. 438. 



290 



ROGERS' DRAWING AND DESIGN. 



J)e5cribit\g C/rc/e 



^pjcycloid 




Fig. 433. "'•«, '^^ 



\ 



■c/ 







ROGERS' DRAWING AND DESIGN. 




291 


i 

i 

1 






rxpxiFh 


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^r- 




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f/ 


O y^ 






"""\//x 




f—\ 




/S. 


/ 






\ \_3t — \ 




P<('"V 


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(\^ 




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) 


cl r~^ 


\i 


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y^^ 1 C 


i 


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c 


h 


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s 


CA. 


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Fig. 434. 




I 


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^ yi 












r\A 










h/\fkj 


^ 





292 



ROGERS' DRAWING AND DESIGN. 



cut the circle A2 by the arc at the point 2 ; this 
point will be the center for the arc of the flank d. 
In this way all flanks may be drawn. 

When the center of an arc, which as closely as 
possible coincides with the face of the tooth ab is 
found, the circle of centers for the faces is drawn, 
and all arcs for the faces of the teeth will lie in this 
circle 

The generating circle may be, within certain 
limits, of any diameter, so long as it is not greater 
than the radius of the wheel on which it is used. 
When the diameter of the generating circle is equal 
to the radius of the pitch circle, the path of any 
point in the circumference of the generating circle 
is a straig^ht line. 

According to the Brown & Sharp system, in cy- 
cloidal gearing, the diameter of the generating circle 
is equal to the radius of a 15-tooth gear of the pitch 
required, this being the base of the system. The 
teeth of the rack of this system have double curves, 
which may be traced by the base circle, rolling al- 
ternately on each side of the pitch line. The same 
generating circle is used for all gears of the same 
pitch. 

According to the prevailing practice, the flank of 
the 15-tooth pinion in cycloidal gearing is made 
radial ; accordingly the diameter of the generating 



circle equals one-half of the pitch diameter of a 15- 
tooth pinion. According to other practice a 12- 
tooth pinion is taken as the base. 

In Fig. 436, is shown a cycloidal rack and pinion. 
The curves of the teeth-profiles for the rack are 
generated by rolling the generating circle along each 
side of the pitch line, on which all pitch points are 
set off. 

A spur gear in which the teeth are on the inside 
of the rim is termed an annular or internal gear 
In such a gear the teeth correspond with the spaces 
of an external gear of the same pitch circle, as do 
also the other proportions of the teeth. They are 
consequently designed in the manner described 
above as involute or as cycloidal gears. 

One particular rule must be observed in regard to 
epicycloidal internal gears ; the difference between 
the diameters of the pitch circles must be at least as 
great as the sum of the diameters of the desci'ibing 
circles. 

Bevel gears are used to connect two shafts which 
intersect when lengthened indefinitely. In most 
cases the shafts are at right angles with each other. 
The pitch surfaces of bevel gears are cones which 
have a common vertex, the point of intersection of 
the axes of the shafts. Fig. 437. 



ROGERS' DRAWING AND DESIGN. 



293 




Fio. 435. 



ROGERS' DRAWING AND DESIGN. 




Before proceeding to draw a pair of bevel gears 
draw a section through the shafts of both gears, 
thus showing a section of one-half of each gear. 
Draw the two axes of the shafts, OA and OB meet- 
ing at O, Fig. 438 shows the two axes at right angles 
with each other. 

Determine the diameter of the largest pitch cir- 
cles in the bevel gears proportionate to the required 
velocity ratio corresponding to the circles which 
form the bases of the two pitch cones. ^ 

Let ef be the maximum pitch diameter of the 
larger, and gh the maximum pitch diameter of the 
smaller bevel gear. An indefinite distance awaj' 
from and parallel to OB draw the line ef ; then draw 
the line gh parallel to OA, each one of these lines 
being bisected by the axes. Through the points e 
and f draw lines parallel to OA and through g and 
h lines parallel to OB. " The lines intersect in the 
points E, F and H. Connect the point O by straight 
lines with the points E, F and H. The resulting 
triangles EOF and FOH are sections of the pitch 
cones of the bevel gear. Make FG equal to the 
width of the face of the teeth. From the point G 
draw the lines IG and GJ parallel to EF and FH 
respectively. Each one of the gears is then com- 
pleted separately with the required proportions for 
the teeth. The manner in which this is done is 



ROGERS' DRAWING AND DESIGN. 



295 




Fio. 438. 



296 



ROGERS' DRAWING AND DESIGN. 



illustrated In Fig. 439. In this figure, ABC is a part 
of the pitch cone laid out according to the principles 
explained with Fig. 438. 

This view must be drawn first. K are the out- 
line's of the teeth of a spur gear laid out for a pitch 
diameter equal to the maximum pitch of the bevel 
gear. The proportions of the teeth on the bevel 
gear are laid off from these outlines. EB and BD 
are the addendum and dedendum ; the line ED 
being drawn at right angles to the line AB of the 
pitch cone. BF is made equal to the length of 
the teeth on the bevel gear ; at F a line perpendic- 
ular to A B is drawn, and the addendum and deden- 
dum of the smallest outline of the tooth is deter- 
mined by the intersection of the line GH with 
the lines EA and DA. Next the other view is 
drawn. 

OJ is a line drawn parallel to BC. A perpen- 
dicular to this line dropped from the point B deter- 
mines the maximum pitch circle and a perpendic- 
ular from the point F the minimum pitch circle. 

In the same manner the addendum and deden- 
dum circles for the largest as well as for the smallest 
profiles of the teeth may be found by dropping per- 
pendiculars from the points E and G, D and H. The 



maximum pitch circle Is then divided Into a number 
of equal parts corresponding to the number of teeth 
required. Through each one of the divisions a line 
is drawn to the center O ; these lines are the center 
lines of the teeth. The proportions of the teeth 
shown at K are then set off from each center line 
for the purpose of forming the projection of the 
teeth. The distance a is set off on the largest de- 
dendum circle, the distance b oA the maximum pitch 
circle ; the distance c on the addendum circle. All 
these lengths are set off so as to be bisected by the 
center line of the tooth for which they are in- 
tended. 

The projection of the smallest profile of the 
tooth Is obtained by drawing radial lines from the 
points at the addendum of the large profile to the 
addendum circle to the smallest addendum cir- 
cle. 

The pitch points of the tooth for the projec- 
tion of the smallest profile at L, is obtained by 
drawing radial lines from the pitch points of the 
large profile to the smallest pitch circle, and the 
root of the smallest profile is obtained in the same 
manner by drawing radial lines from the dedendum 
points of the large profile to the smallest dedendum 
circle. 









ROGERS' DRAWING AND DESIGN, 




297 










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Fig. 439. ^^^^^^^=5:&/ 







298 



ROGERS' DRAWING AND DESIGN. 



In speaking of the diameter of a bevel gear, the 
largest diameter of the pitch cone is meant. 

The relations between the pitch diameter of a 
bevel gear pitch, number of teeth and velocities are 
the same as for spur gears, and all calculations are 
made in the'same manner. 

In Figs. 440, 441 and 442 are shown 3 views of a 
completed bevel gear. 

For transmitting motion from one shaft to another 
at riofht anofles to it, when the axes of the shaft do not 
intersect, the worm gear and worm shown in Figs. 
443 and 444 are used. The section of the worm 
shown in Fig. 445 is of the same outlines as a rack 
of corresponding pitch, and may be of either the 
involute or cycloidal form. The involute form is 
generally adopted, as its teeth are easier to pro- 
duce. The worm is cut in a lathe like a screw. 
The diameter of the worm is ordinarily taken about 
5 times the pitch, although it could be made of any 
convenient diameter. The worm must always be 
the driver. It is not well adapted to the transmis- 
sion of heavy power, as the tooth action is purely a 
slidingf one. 

Fig. 446 shows a partial section of a worm gear 
made by a plane perpendicular to the axis of the 
worm through the axis of the gear. 



When made by the involute system the worm 
teeth will be straight. The worm may be drawn 
by the aid of helices like a screw. The worm may 
also be single, double, etc., like a screw. 

Figs. 443 and 444 show the outside views of a 
worm gear. Fig. 446 also shows tl)e outside view 
of a complete tooth. 

The drawing of the outside views of 'i worm gear 
involves considerable labor. Fortunately, however, 
it is wholly unnecessary for the purposes of machine 
; shop construction, to make complete outside views 
of worm gears. The same is also true in the case 
of bevel gears. A sectional view is most generally 
adopted to show the wheel. The wheel is usually 
made to embrace about one-sixth the circumference 
of the worm. 

vVhen the diameter of the worm is increased and 
approaches the diameter of the wheel and when the 
worm is given a multiple thread and the number of 
teeth in the worm wheel is comparatively low, then 
both worm and worm wheel take the shape of spiral 
gears. 

To add strength to spur gears, the rim is made 
wider than the teeth, and is carried outward, as 
shown in Figs. 447 and 448. This is called shroud- 
ing of the teeth. If two wheels gearing together 
do not differ greatly in diameter, each may be 



ROGERS' DRAWING AND DESIGN. 



299 




Fig. 440. 



Fig. UL 



Fig. 442. 



300 



ROGERS' DRAWING AND DESIGN. 



shrouded to the pitch on both sides ; but when one 
is very much larger than the other, it is usual to 
shroud the smaller only. 

For light spur gears the rim is generally made as 
shown in Fig. 449. The section shown in Fig. 450 
shows the rim of a heavy spur gear. The propor- 
tions marked on the sections in these figures are in 
terms of the circular pitch of the gears. 

Sections of arms for gear wheels are shown in 
Figs. 451, 452 and 453. For light spur gears the 
section shown in Fig. 451 is used. For heavy spur 
wheels the sections shown in Fig. 452 is adopted. 
The section in Fig. 453 is for bevel gears. The 

number of arms is approximately + 4 when 

D is the diameter of the pitch circle in inches. 
The nearest even number may be taken. 

The width of the arms may be calculated in the 
following manner. In the arm shown in Fig. 451 
the greatest breadth of the arm is equal to B ; then, 

1.6 X width of tooth X cir- 

cular pitch X pitch diam. 

2 X number of teeth 

and for the other sections, 

T53 width of tooth X pitch diameter 
2 X number of teeth. 



A practical rule for finding the thickness of the 
hub is 



The thicknes of hub = 



1.6 X circular pitch X 0.2 
pitch diameter 



The length of the hub is, in most cases, equal to 
the width of the tooth ; it may vary up to i}4 times 
the width of the tooth. 

The pressure on one tooth may be taken, for mqat- 
purposes, to equal | of the whole pressure ; that is 
of the driving force at the pitch line, calculated from 
the horse power and the speed. 

H X 33,000 



P = 



V 



where H is the given horse power and V the veloc- 
ity of the spur gear. The velocity of the given 
wheel is found by multiplying the circumference in 
feet by the number of revolutions per minute. The 
thickness of the tooth on the pitch line may be found 
from the following formula, ?'. c, 



Square of thickness= 



3 times pressure on one tooth 
safe stress 



For cast iron gears the safe stress may be taken 
to equal 4,000 pounds per sq. in. 



ROGERS' DRAWING AND DESIGN. 



301 




Fia. 443. 



Fia.444, 




302 



ROGERS' DRAWING AND DESIGN. 




ROGERS' DRAWING AND DESIGN. 



303 



FlO. Hi. 



f 

047 




^M 



FiQ.aa. 







Fig. 4i8. 




ITiO. 450. 




Fig. 451. 



rS 






^ — ■*- 



Fig. 452. 



-B 






1 



Fig. 453. 



304 



ROGERS' DRAWING AND DESIGN. 




Fig. 454. 



Example : 

If the whole driving force at the pitch circle is 
equal to 12,000 pounds, then the pressure on one 
tooth will be 8, coo pounds, and the square of the 

1-1 r 1 1-11 1 8,000 X 3 

thickness 01 the tooth wul equal =6. 

^ 4,000 

The thickness of the tooth must then be equal to 

the square root of 6, i. e., 2.449 •"> ^^Y '^% \nc\\ 



TRAINS OF GEAR WPiEELS. 

When a train of gear wheels is employed in a 
machine, the usual arrangement is to fasten two 
gear wheels of unequal size upon every axis, except 
the first and the last, and to make the larger wheel 
of any pair engage the smaller one of the next pair. 

If the wheel A in Fig. 454 is the driving wheel, 
and the wheel F the last follower, and if it be deter- 
mined that for each single revolution of A, the wheel 



F shall make 50 revolutions, then it is said that the 
value of the train is equal to 50. 

That is, the value of a train of gear wheels is 
equal to the number of revolutions in the last fol- 
lower in a given time, divided by the number of 
revolutions of the main driver in the same time. 

Suppose that the first wheel had A teeth, the 
second B, the third C teeth, the fourth wheel D 
teeth, the next E and the last F teeth ; then, 

-^-== the velocity ratio between the first and the 
B 

second axes upon which are fastened the wheels B 

and C ; 

-— = the velocity ratio between the second and 

third axes ; 

JE 

T 
and 

A 
B 



^^ velocity ratio between third and fourth axes, 



X- 



C 



X 



,^ == the velocity ratio between the 
D h 

two extreme axes, that is, it will equal the value of 

the train. 

Example : 

Let A have 120 teeth, B 15 teeth, C 100 teeth, 
D 50 teeth, E 150 teeth and F 30 teeth. 

Then, the value of the train is equal to 



120 
15 



-X- 



100 

50 



X 



150 



80 



METAL WORKING MACHINERY. 



The discovery of metals and the means of working them are among the first stages in the develop- 
ment of primeval man ; the earliest evidence of a knowledge and use of metals is found in the primitive 
implements of the so-called Bronze and Iron Age. Attention is called to the interesting note below. 

The Old Testament mentions six metals — gold, silver, copper, iron, tin and lead ; the old Greeks in 
addition to these, and to bronze, came also to know mercury ; the same set of metals without addi- 
tions seem to be the only ones known until the Fifteenth Century when atitimony was discovered ; about 
1730 A. D., arsenic and cobalt were discovered, nickel and manganese were, discovered in 1774; in the 
meantime something had become known in a general way of zinc, bismuth diXxA platinum. 

Since the date last mentioned the discovery of many rare metals has become frequent, aluminum 
being among the last most useful and interesting discoveries of metals unknown at the beginning of the 
Nineteenth Century. 

The following pages deal, in text and illustrations, with iron working machinery, as agamst those 
machines devised to work i7i luood, etc., and few as are the cases named they show vividly the progress 
made in the methods of working the metals named. 

In designitig machines it is well to keep in mind, i, that each machine ought to be made of as lew 
parts as possible, 2, as simple as possible, 3, the strength of every part should be made proportional to 
the stress it has to bear, 4, all superfluous weight which clogs the machine's motion should be avoided, 
5, all parts should be contrived to last equally well, 6, in wheels with teeth, the number of teeth that 
play together ought to be so constructed that the same teeth may not meet at every revolution, but as 
seldom as possible. 

Note. — " Some recent analyses of the iron of prehistoric weapons have brought to light the interesting fact that many of the prehistoric 
specimens of iron manufacture contain n consiilerable percenlage of nickel. This special alloy does not occur in any known iron ores but is 
invariably found in meteoric iron. It thus appears that iron was manufactured from meteorolites which had fallen to the earth in an almost 
pure metallic state, possibly long before prehistoric man had learned how to dig for and smelt iron in any of the forms of ore which are found 
on this planet." — Enc. Briiannica. 

307 



308 



ROGERS' DRAWING AND DESIGN. 



DES AND PRESSES. 

The use of dies and presses has increased in 
recent years to an almost marvelous extent, and a 
numberless variety of articles are now being pressed 
out easily and rapidly by the aid of dies, which in 
former times involved great labor as well as a long 
special training. The number and variety of dies 
are so very large that it is beyond the limits of this 
book to give even a partial list or classification of 
these useful tools. 

In this section we shall limit ourselves to a few 
examples of the most frequently employed forms of 
dies, so as to give the reader an opportunity to ac- 
quaint himself with this importamt part of modern 
mechanism sufficiently to understand further special 
literature upon this subject, if it should be the 
desire of the reader to make a thorough study of 
this branch of machine shop practice. 

The simplest form of a die is a blanking die. 
Blanking dies are made for the purpose of cutting 
out various pieces of metal from a comparatively 
thin sheet of metal, cardboard, etc., leaving the cut 
out piece perfectly flat ; this piece is called a blank. 

A set of blanking dies consists of a male die, or 
punch, and the lower or female die. The lower die 



has an opening exactly equal to the form of the 
punch. 

Fig. 455 shows a punch and die for a circular 
blank. The narrow part of the punch, the shank, 
is preferably made in one piece with the punch. 








Fig. 455. 



The shank is fastened to the ram of the press, while 
the die is secured to the bed of the press. The 
taper of the lower die gives the clearance, required 
for the purpose of facilitating the dropping of the 
blank from the die as soon as it is cut ; the clear- 



ance is made from 



to 3' 



for average work. 



ROGERS' DRAWING AND DESIGN. 



309 



For the purpose of greater rapidity of work and 
uniformity in the matter of spacing the holes, dies 
and. punches are grouped; that is, several punches 




Fig. 456. 



are fastened to one shank, while several separate 
openings are worked out in the lower die to corre- 
spond with the number of required separate dies, if 
the work were done by single dies. 





V 



Fio. 457 



Fig. 458 shows such a gang die, as it is called, 
made for cutting out washers of the shape shown in 
Fig. 459. One stroke of the die produces two holes, 
as shown in Fig. 4^7. The metal is fed into the 




Fig. 458. 




KiG. 459. 



310 



ROGERS' DRAWING AND DESIGN. 



die from the side of the smaller hole. The small 
punch will cut a hole in it equal to the inside diam- 
eter of the washer, as shown in Fig. 456 ; the metal 
is then advanced and at the second stroke the large 
punch will cut out the complete washer, while the 
small punch pierces the metal for the next washer at 
the same stroke. The plate S, Fig. 458, is the 
stripper which takes the metal off the punches on 




Fig. 460. 



their upward stroke. It is evident that the metal 
must be fed below the stripper. 

Fig. 461 shows the simplest form of a drawing 
punch and die. The flat circular blank. Fig. 460, 
is placed upon the die so as to fit the set edge S, 
and is pushed through the die by the punch. While 
the punch returns upward, the finished shell is pulled 



off by the edge P, which is made very sharp for that 
purpose. The diameter of the punch is equal to 
the diameter of the die, minus two times the thick- 
ness of the blank. 




Fig. 461. 



Fig. 462 shows the shell. This simple form of a 
drawing die should be used only on shallow work, 
to avoid crimping around the edge of the shell. 
When the blank is held firmly while being drawn, 
the crimping even on deeper work may be avoided. 



ROGERS' DRAWING AND DESIGN. 



311 





O 



(O; 



tr 



%J' 



I 

w 



tr 



r^ 



\^ 



Fig. 466. 



Fig. 463. 



Fio. 464. 



Fig. 465. 



312 



ROGERS' DRAWING AND DESIGN. 




Another gang die is shown in Figs. 463 and 465. 

Fig. 466 is the blank, Fig. 465 the top view of the 
lower die and stripper ; Fig. 464, a sectional view of 
the same ; and Fig. 463 shows the punch in section. 




Fig. 469. 



T^G. 468. 



In Fig. 468 is shown a type of a die, which by its 
relative simplicity when compared with the work 
produced, will always stand as a beautiful example 
of mechanical ingenuity. 



ROGERS' DRAWING AND DESIGN. 



313 



It is a single-action cutting and drawing die, gen- 
erall)- called a single-action combination die; it is a 
combination of a blanking and a drawing die in one ; 
it cuts the blank and draws it up into the shell shown 
in Fig. 467 at the same stroke. In descending the 
blank is cut by the edge of the blanking punch B, 
meetinor the edg-e of the blanking die A. The blank 



continues until the drawing punch C is drawn down 
on the drawing die D, when the blank is drawn into 
the required shape. 

Fig. 469 shows a sheet iron dynamo armature disc. 
Fig. 470 is a section on a larger scale of a set of 
dies of latest construction, designed for cutting such 
discs. These dies are made to cut discs up to 100" 




Fig. 470. 



is then held firmly by the blank holder ring E and 
is forced down together with it, by continued down- 
ward motion of the blanking punch B. The blank 
holder ring is forced up by the elastic force of the rub- 
ber spring barrel R upon which the ring sets, through 
the medium of six pins passing through the bolster or 
die holder. The descent of the blanking punch B 



in diameter, and it is claimed that when the press is 
run at a speed of 55 revolutions per minute, nearly 
6,000 sheets 20" in diameter may be produced in ten 
hours. 

Fig. 471 shows a well-known type of punching and 
shearing machine. It will be noticed that the ma- 
chine is powerfully geared. The machine is really 



314 



ROGERS' DRAWING AND DESIGN. 



a form of a press and contains all essential parts of 
such a mechanism. 

It will also be noticed that the machine is equipped 
with a stop clutch, operated by a foot-lever. The func- 
tion of a stop-clutch is, at the will of the operator, 
to suddenly make a driving connection between the 
constantly revolving gear wheel and the tempor- 
ary stationary main shaft. Its further purpose is to 
disconnect these members again automatically, after 
the shaft has made exactly one revolution and when 
the punch has reached its highest open position. 



DRILLING MACHINES. 

A mechanism of the greatest importance in a 
machine shop is the drilling machine. The ordinary 
drill press, as the larger drilling machines are gener- 
ally called, is a complicated machine tool, presenting 
a great number of interesting mechanical principles 
to the student. It is, however, not within the scope 
of this book to take up extensively the construction 
of this machine ; the figure on page 317 is an exam- 
ple of this class of machinery ; the drawing shows 
a bench drill, which embodying, as it does, all the 
parts essential to any drilling machine, will enable 
the student to understand the favorite types of the 
drilling machine. 



The following is a table of the speeds of 
drills for different sizes of drills and for different 
metals, as recommended by the Cleveland Twist 
Drill Company : 

Table of Drill Speeds. 



Diam- 
eter of 
Drill. 


Speed 

for Soft 

Steel. 


Speed 

for Cast 

Iron. 


Speed 

for 
Brass. 


Diam- 
eter of 
Drill. 


Speed 

for Soft 

Steel. 


Speed 

for Cast 

Iron. 


Speed 

for 
Brass. 


1 


1,824 


2,128 


3,648 


ItV 


108 


125 


"215 


* 


912 


1,064 


1,824 


H 


102 


118 


203 


A 


608 


710 


1,216 


lA 


96 


112 


192 


1 


456 


532 


912 


li 


91 


106 


182 


5 


365 


425 


730 


ItV 


87 


101 


174 


1- 


304 


355 


608 


If 


83 


97 


165 


t'h 


260 


304 


520 


lA 


80 


93 


159 


1 


228 


266 


456 


U 


76 


89 


152 


tV 

5 


203 
182 


236 
213 


405 
365 


lA 


78 
70 


85 
82 


146 

140 


w 


166 


194 


332 


^\ 


68 


79 


135 


f 


152 


177 


304 


If 


65 


76 


130 


1 3 
fF 


140 


164 


280 


m 


63 


73 


125 


I 


130 


152 


260 


n 


60 


71 


122 


If 


122 


142 


243 


in 


59 


69 


118 


1 


114 


133 


228 


2 


57 


67 


114 



ROGERS' DRAWING AND DESIGN. 




316 



ROGERS' DRAWING AND DESIGN, 



Fig. 472 shows a side elevation of a bench drill 
of a neat and practical design, suitable for drilling 
small holes. A front view of this machine is illus- 
trated in Fig. 473, while Fig. 474 exhibits a top 
view of the same. 

The principal parts of the machine are the vertical 
spindle, holding the drilling tool, a table upon which 
the work to be drilled is held, and a rigid frame to 
which all parts of the machine are fastened. 

The driving cone pulley, as well as the fast and 
loose pulley are mounted upon one horizontal shaft, 
at the back of the frame near its lower end. 

The driving belt is led upward from the cone 
pulley over two horizontal guide pulleys, and then 
in a horizontal direction to the cone pulley, which 
is mounted upon the vertical spindle near its highest 
point. 

The lower end of the spindle is provided with a 
thread for the purpose of holding a chuck which is 
to receive the drilling tool. The weight of drill, 
chuck, spindle and other parts which move down 
ward or upward together with these parts are 
counterbalanced by a weight which is hidden in the 
hollow frame. For the upward and downward 
motion of the spindle a pinion and rack motion is 
provided. The table m.iy also be lowered or raised 
according to the requirement of the work. 



The number of varieties of drilling machines is 
growing rapidly. Large drilling machines are used 
also for tapping holes and are generally provided 
with automatic feed. 

A class of drill presses which is particularly 
adapted for larger work is known as the radial drill ; 
in this type the work is not shifted, after drilling a 
hole, if there are more holes to be drilled into the 
same surface ; the drill spindle, with its entire 
mechanism is mounted upon a heavy cast iron ai^m, 
which swings horizontally upon the frame of the 
machine ; the arm may be lowered or raised to suit 
the work, and the spindle carriage can be moved in 
or out on the arm, to suit conditions. 



THE MILLING MACHINE. 

The Milling Machine may be classed as a combi- 
nation of several other machine tools used for cutting 
metals ; the work that can be done on this machine 
is not limited to either straight or curved surfaces, 
or drilling of holes ; in general construction this 
machine does not differ greatly from an ordinary 
drill press, in fact it can often be used in its place. 

In this machine the table upon which the work is 
held is movable in all directions, without disturbing 



ROGERS' DRAWING AND DESIGN. 



317 





t 3ji' i 




Fio. 474. 



Fig. 472. 



Fig. 473. 



318 



ROGERS' DRAWING AND DESIGN. 



the adjustment of the work, which is fed either auto- 
matically or by hand feed, while the rotating cutter 
removes the superfluous metal. 

In illustrations Nos. 475 and 476 are illustrated 
an approved form of a vertical spindle milling 
machine. 

Fig. 475 shows the front view of this machine, 
and Fig. 476 shows the side elevation. The whole 
machine is an advanced type of a modern milling 
machine and produces an impression of strength 
and neatness of design. 

The vertical spindle of this machine is made 
fully 3" in diameter ; the lower end of the spindle is 
provided with a thread for large mills, working in a 
horizontal plane. 

77^1? platen as well as the saddle of this machine 
are 5 1]/^' long. All feed screws are provided with 
dials, thus enabling accurate work in a most conven- 
ient manner. 

The largest distance between the spindle and the 
platen is 2ii/^". The extreme distance between 
the rotary table and the vertical spindle is 16". 
There are fully eight changes of feed for the table 
and sixteen changes for the rotary attachment. 

The dimensions of this machine over all, are as 
follows: Height 81", width 65" and depth 88}4"- 




ROGERS' DRAWING AND DESIGN. 



319 



a 




Fig. 476. 



320 



ROGERS' DRAWING AND DESIGN. 




i^S 




> 

^ 



rsr 




\ 



-en 



Fig. 4TT. 




FlO. 478, 




DRAWING AND DESIGN. 



321 



THE LATHE. 

The most important machine tool in a shop is 
undoubtedly the lathe. It is used for a great variety 
of purposes and for this reason it is made in many 
different special forms and designs. 

The simplest kind of a lathe is shown in Figs. 477 
to 479. It is called a speed lathe and is used for 
small work which can be run at a high speed. 

The lathe is composed of the following principal 
parts : 

The bed. 

The legs or supports. 

The head stock. 

The tail stock. 

The tool rest. 



I. 
2. 

3- 
4- 

5- 



By means of the steps in the cofie pulley on the 
head stock different changes of speed of the spindle 
can be obtained. The tool rest, V\^^. \%o and 481, 
is adjustable in all directions, but it is not provided 
with automatic feed connections. 

The ordinary engine lathe, used for heavier and 
more accurate work, has the same main parts as the 
speed lathe. In this lathe, however, the carriage 
with its tool support is moved over the shears of the 
bed by the lead screw and its connections. The lead 
screw is splined and the feed mechanism is driven 



322 



ROGERS' DRAWING AND DESIGN. 



from a collar which has 2^. feather engaging the spline 
and slides over the lead screw. The form of thread 
used on lead screws is somewhat similar to a square 
thread with sides forming an angle of 14^ degrees. 

The lead screw is driven from the spindle of the 
head stock by gear wheel connections. 

The head stock of an engine lathe, in two views, is 
shown in Figs. 482 and 483. Fig. 482 shows an ele- 
vation of the head stock, and Fig. 483 represents its 
plan or top view, the back gears being plainly shown. 

Large engine lathes are also provided with a 
separate feed shaft besides the lead screw ; this 
shaft is driven by a belt and cone pulleys, from 
the stud, and is splined lengthwise ; a splined worm 
is fitted upon this shaft in such a manner that 
it can slide on it lengthwise, but is held by two 
projections on the apron of the carriage, so that it 
will slide with the carriage and at the same time 
turn with the feed shaft. 

This worm engages in a worm-wheel, connected 
by a clutch to a gear, which meshes with a rack 
under the front edge of the lathe-bed. By means 
of this clutch the feed can be engaged or disen- 
gaged. The worm-wheel also connects with a 
clutch, which will operate the cross-feed of the tool. 
Both clutches are operated by knobs at the front 
of the apron. 



Fig. 484 shows the longitudinal section and Fig. 
485 is a cross or lateral section of the tail-stock of a 
lathe of the usual form. All back-geared lathes can 
be run with the back gears idle, by locking the cone 
with the spindle gear; in this way the spindle has 
only the changes in spindle speed depending upon 
the steps of the cone pulley as in the speed lathe. 
The gear wheels at the back of the head stock reduce 
the speed of the spindle and a double number of 
changes in speed can thus be obtained. 

Figs. 486 and 487 show the mannfer in which the 
spindle gear may be connected with the lead screw 
gear, for producing different feeds. 



CHANGING GEARS FOR SCREW CUTTING. 

The problem of cutting a screw on a lathe resolves 
itself into connecting the spindle of the lathe with 
the lead screw by a number of gears in such a man- 
ner that the carriage, moved by the lead screw, 
advances exactly one inch during the lapse of time 
required for the lathe spindle to make a number of 
revolutions equal to the number of threads to the 
inch in the desired screw. 

The lead screw has, nearly always, a single 
thread, and, therefore, to move the carriage for- 



ROGERS" DRAWING AND DESIGN. 



323 




CenferLine 



3" t-d.m S/ideTTest- 



I 



Ufi 



±^- 





Fig. 481. 



324 



ROGERS' DRAWING AND DESIGN. 




Fto. 482. 



ROGERS' DRAWING AND DESIGN. 



325 




Fio. 483. 



326 



ROGERS' DRAWING AND DESIGN. 



ward just one inch it must make a number of 
revolutions equal to its own number of threads per 
inch. It is, consequently, first of all necessary to 
know the number of threads per inch on the lead 
screw. 

The spindle of the lathe is provided with a gear 
which transmits the rotary motion of the spindle to 
the stud gear, below the spindle, by means of inter- 
mediate gears, situated within the head stock. There 
are two of these intermediate gears, one being an 
idle gear, for the purpose of changing the direction 
of the motion of the stud and through this the lead 
screw. 

The connection of the stud with the lead screw 
may be accomplished by simple or cotnpoujid gearing. 

In simple gearing the motion of the stud gear is 
transmitted either direct or by means of an inter- 
mediate gear to the gear on the lead screw. One 
or more intermediate gears, which simply transmit 
the motion received from one gear to another, do 
not affect the resulting ratio of a train of gears. 
Consequently, the intermediate gears in simple gear- 
ing will be disregarded in all calculations for screw 
cutting. 

The stud gear is usually equal to the driving gear 
on the spindle; it may, however, be of a different 
size and in the following problem it will be assumed 



that the gear on the spindle has double the number 
of teeth than that on the stud. 

The following formula will give the required ratio 
for the gears on the stud and on the lead screw : 

Number of teeth on stud gear 
Divided by number of teeth on lead screw gear 



Number of turns of spindle 



multipled 



by 



Number of turns of stud 

Number of threads on the lead screw 



Divided by number of threads per inch 
on required screw. 

Problem : 

It is required to cut a screw with i6 threads to 
the inch ; the lead screw has 8 threads to the inch 
and the spindle makes 20 turns to 40 turns of the 
stud. 



Solution : 

Number of teeth on stud gear 20 8 i 

Number of teeth on lead screw gear 40 16 4 

The required ratio is one to four, i.e., when the 
stud gear will have 16 teeth the lead screw gear will 
have 16 X 4^64 teeth; now if the stud gear will 
have 20 teeth the lead screw gear will have 20X4 = 
80 teeth and so on. 



ROGERS' DRAWING AND DESIGN. 



327 




■,■ •.ftfN}}WW?^.V/?f 



Fig. 484. 




328 



ROGERS' DRAWING AND DESIGN. 



In compound gearing, as in Fig. 
487, the motion of the stud gear 
is transmitted to the lead screw 
by two gears keyed together on 
an intermediate stud. In this 
case there are four changeable 
gears and consequently a wider 
range of changes than in simple 
gearing. 

Of the two gears working to- 
gether on the intermediate stud 
that one which works with the 
spindle stud is called the first gear 
and the other working with the 
lead screw g-ear is termed the sec- 
ond gear. 

Now assuming that the spindle 
gear makes 20 revolutions to 40 
revolutions of the stud gear and 
that the lead screw has 4 threads 
to the inch it will be necessary to 
find the velocity ratio between the 
stud gear and the lead screw gear 
for cutting a screw with 50 threads 
to the inch. 




Fig. 



^imhla Qearing. 



ROGERS' DRAWING AND DESIGN. 



329 




Number of teeth in stud 
_a;ear 



20 4 
40^50 



I 
25 



Fio. 487. 



Comhound (^eartny. 



Number of teeth in lead 
screw gear 

that is, if the gear on the stud should have 
16 teeth then tlie gear on the lead screw 
would have 400 teeth, or the required 
ratio for simple gearing. For compound 
gearing this ratio can be divided into fac- 
tors, for instance, iX|=5V' that is the 
velocity ratio for the spindle stud gear and 
the first intermediate stud gear could be 
made equal to i, and the same velocity 
ratio for the two other gears. For in- 
stance, if the stud gear will have 16 teeth 
the first intermediate stud gear must have 
5 X 16^80 teeth; the second intermedi- 
ate stud gear could have 16 teeth and the 
lead screw gear 80 teeth. Or 15 and 75 
could be taken for the first pair and 16 and 
80 for the second pair, or in fact any pair of 
gears having the desired velocity ratio. 

Figure 488 represents a modern shaft- 
ing lathe, built by the Springfield Tool 
Company, and which can be used both for 
ordinary lathe work and especially for 
turning shafting. 



330 



ROGERS' DRAWING AND DESIGN. 



m im I ffitf rm fq| 



£ 





(ml 




■Q_s z T 



rk. 




Fig. 488a. 



ROGERS' DRAWING AND DESIGN. 



331 






no. laaB. 



Fig, 489. 



Fig. 490. 



332 



ROGERS' DRAWING AND DESIGN. 




This lathe may be fed by a friction feed or by 
the lead screw. Slightly below and midway between 
the shears in the bed, passes a splined shaft, which 
drives the tail stock face plate by means of an inter- 
mediate gear, shown in Fig. 490. This small gear 
can be thrown out of action by the worm-wheel sec- 
tion shown in Fig. 489. The tail stock face plate 
is made for the purpose of driving the shaft by a 
dog, exactly as is done by the head stock. It is very 
convenient for turning the end of the shaft for which 
the head stock dog has to be removed. 

The long centers are a necessity in this lathe in 
view of the fact that they must reach through a 
bushing in the rest. This bushing is depended upon 
to support the shaft during the cutting ; it is made 
to fit the shaft exactly. The special rest for shaft- 
ing slides into place on the saddle when the com- 
pound rest is removed. The pump and tanks for 
lubricants are carried by this rest. The pump is 
driven by a gear wheel seen in Fig. 491, which 
engages with a pinion sliding upon a shaft ; this 
latter extends the entire length at the back of the 
lathe. Fig. 491 also shows the lower tank into which 
all the lubricant collects, from where it flows by 
gravity to the pump. 



ENGINES AND BOILERS. 



The study of the steam engine involves an acquaintance with the sciences of heat, of chemistry, and 
of pure and applied mechanics, as well as a knowledge of the theory of mechanism and the strength of 
materials ; many other things are needed to be known, as the student will find as he progresses in his 
researches, the first of which should relate to the safe and economical production of the steam itself. 

Nearly the whole of the Eighteenth Century passed in experiments made to reduce the energy, 
latent in coal and other fuels, to the service of mankind ; at its earliest point of progression the boiler 
and the engine were substantially one and the combined engine and boiler were known as the fire engine. 
At a little later period when scientific research had shown clearly the source of the power which gave 
vitality to the newly invented mechanism the name changed to the heat engine, it having become known 
that heat accomplishes work only by being let down from a higher to a lower temperature, a certain 
amount of heat disappearing when changed into work. 

The modern Steam Engine is now considered as apart from the Steam Boiler and the classification 
and variety of each and the successive steps of advancement, while full of interest are too voluminous to 
consider in this volume, but some account of their early history is given in the note below. 

Note. — About the year 1710, Thomas Newcomen, ironmonger, and John Cawley, glazier, of Dartmouth, in the county of Devonshire, 
made several experiments in private, and in the year 1712 put up an engine, operated by steam, which acted successfully. The progre.ss made 
was very rapid and it is recorded that in the year 1737 there was a pumping engine of the Newcomen construction working a succession of 
pumps each 7 inches in diameter and 24 feet apart, and making 6-feet strokes at the rate of 15 per minute, whereby water was pumped from cis- 
tern to cistern throughout the whole length of a shaft 267 feet deep, by steam at or near the atmospheric pressure. 

The construction of the Newcomen engine was greatly improved by Smeaton, who designed and erected an engine for the Chase-Water 
mine, in Cornwall, which had a cylinder of 72 inches in diameter, with a 9-feet stroke, and worked up to 76 horse power. There were three 
boilers, each fifteen feet in diameter. This was the last effort on a system then about to pass away, for the engine was set up in 1775 no less 
than six years after the date of Watts' patent, and we are told that when " erected it was the most powerful machine in existence." 



336 



ROGERS' DRAWING AND DESIGN. 



STEAM BOILERS. 

A closed vessel in which water may be heated for 
the purpose of generating steam is called a steam 
boiler ; the boiler is partially filled with water for 
this purpose, the level of the water in the boiler be- 
ing called its water line ; the space above the water 
line is termed the steam space. 

That part of the surface of the boiler, exposed to 
the heat of the fire and of the hot gases, is called 
the heating surface of the boiler, and its measure- 
ment is usually given in square feet. 

The space in which the heat is generated is called 
the furnace ; the surface upon which the coal is laid 
is called the grate surface, and its dimensions are 
also given in square feet. 

Steam boilers may be classified according to their 
construction and form, or according to their appli- 
cation. Thus, we have horizontal and vertical boilers, 
externally and internally fired boilers, plain cylin- 
drical shell boilers, fire-tube and water-tube boilers. 

Boilers may be stationary o'c portable ; there are 
locomotive boilers and marine boilers semi-portable, 
etc. 

The plain cylindrical boiler, Fig. 492, consists of 
a long cylinder called a shell, made of iron or steel 



plates riveted together, the ends of the cylinder be- 
ing closed by flat plates called the heads of the boiler. 

The furnace is arranged at the front end of the 
boiler, the fuel being placed on the grate through 
the furnace door, the ashes falling through the 
grate into the ash pit below. Behind the furnace 
is built a brick wall, called the bridge wall, which 
keeps the hot gases in close contact with the under- 
side of the boiler. ^ 

For long boilers of this class a second bridge-wall 
often is built. 

The hot gases flow from the furnace over the 
bridge walls into the chim,ney. Within the chimney 
is placed a damper regulating the flow of these gases. 
All portions of the brick work exposed to the action 
of the gases, are made of fire-brick. It is not desir- 
able to allow the upper portion of the boiler to come 
in contact with the hot gases, and for this reason the 
boiler above the water line is lined with fire brick. 

Water is forced into the boiler through the feed 
pipe leading to the lower end of the boiler, by the 
aid of an injector or a pump. To prevent the steam 
from rising above a certain pressure, a safety valve 
is placed at the top of the boiler. 

From the highest part of the boiler also, the main 
steam pipe leads the steam to the engine or any 
apparatus for which the steam is to be used. 



ROGERS' DRAWING AND DESIGN. 



337 



V//^^/////////////////////7^^ '-'- '^^'':^:- ^-^X' : '^< t'. '-'<■ <y-:^ ^ 







Fig. 493. 



388 



ROSERS' DRAWING AND DESIGN. 



The pressure of the steam in the boilers is indi- 
cated by /"/z^ i'/!'^?^^ ^azit^^, Figs. 493 and 494, attached 
to a pipe which passes through the front head into 
the steam space of the boiler. To determine the water 
level, gauge cocks are placed in the front head of the 
boiler shell. There are, commonly, three of them 




Fig. 493. 



Fig. 491. 



in one vertical line ; the level of the water may be 
found approximately by opening these gauge cocks, 
shown in Fig. 495. 

The level of the water may also be shown by a 
water glass, illustrated in Fig. 496. Valves at the 
top and bottom.allow the steam to be shut off if the 
glass breaks or needs cleaning, and 2i pet cock at the 



d 



To SfeamSfjac^ 



Gauge Glasb 







I^ater6pace 



cie: 



-^ Top of l/^Joer Tfou/ 
of Tubes 



^^To Ashpit 



Fig. 495. 



ROGERS' DRAWING AND DESIGN. 



339 



bottom allows the water to be blown out of it. The 
brass fittingrs of the water glass are often screwed 
directly into the boiler plates ; when this cannot be 
done, the water glass may be put on a water column 
like that shown in Fig. 495. 

The boiler must also be provided with a blozu-off 
pipe, through which the water may be discharged ; 
in Fig. 492 the feed pipe, as well as the blow-off 
may be seen at the rear of the boiler. 



A nianhole,¥\g. ^g/, is constructed in the front head 
or on top of the boiler, to allow a person to enter 
for the purpose of cleaning, inspecting or repairing. 
At the lower side of the boiler a handhole is gener- 
ally sufficient, as in Fig. 498, for cleaning it out and 
removing the accumulated sediment. 





Fig. 496. 



y//////////^. 




\\\\V\\\\\\^\\\\\\\\ ^L^^^^^ 




FIQ.4W. 



w/yyyy/yyyyy. 



340 



ROGERS' DRAWING AND DESIGN. 



a 



« 



ii_i 




Fig. 498. 



Plain cylindrical, or plain shell boilers, may be 
made from 30 to 40 inches in diameter and from 20 
to 40 feet lono^ ; they are not considered economical 
on account of their small heating surface. 

The jiue boiler. Figs. 499 and 500, differs from 
the plain shell boiler in having one or more large 
flues running lengthwise through the shell, below 
the water level. The hot gases pass from the fur- 
nace over the bridge walls and then through the 
flues to the smoke box and chimney. This type of 
boiler is more economical than the plain shell boiler, 
as the flues considerably increase the heating sur- 
face. It is used wherever the water is bad, even 
though it may not be as economical as some other 
types of boiler. 

The cylindrical tubular boiler. Figs. 501, 502, 503 
and 504, is a development of the flue type. It con- 
sists of a cylindrical shell, closed at the ends by two 
flat tube plates and of numerous fire-tubes, usu- 



ally having a diameter of three or four inches. 
About two-thirds of the boiler is filled with water ; 
the other third being the steam space. The water 
level must be six or eight inches above the highest 
row of fire-tubes. 

The fii-e-tubes hold the two ends of the boiler 
rigidly together, acting as stays, but as these are 
placed only below the water line the upper parts of 
the flat plates are braced by through rods or stays, 
or diagonal stays, similar to that shown in Fig. 505. 

The boiler is supported by the side walls by means 
of brackets riveted to the shell, similar to the one 
shown in Fig. 506. 

The Cornish boiler consists of a cylindrical shell 
with flat ends, through which passes a smaller tube 
or flue, containing the furnace as shown in Fig. 
507. The furnace is terminated by the bridge wall 
built of fire brick. The hot gases flow over the 
bridge wall to the end of the furnace tube, then they 



ROGERS' DRAWING AND DESIGN. 



341 



return on both sides of the boiler shell through the 
flues on the side of the boiler and again pass to the 
back end of the boiler by the flue running along its 
bottom to the chimney. The figure shows a longi- 




FlG. 199. 



tudinal section of the boiler. The gases part with 
much of their heat before reaching the bottom of 
the boiler, and therefore are less liable to unduly 
heat the bottom plates where sediment usually col- 



lects. It has been found that this method of draft 
heated the plates unequally and thus weakened the 
boiler. For this reason the products of combustion 
are led through the bottom flue first and then only 




Fig. 500. 



through the side flues, this method being called the 
split draft. 

A Cornish boiler having two internal furnace tubes 
is called a Lancashire boiler. 



342 



ROGERS' DRAWING AND DESIGN. 



^— 5r — b ° ° " 5 — -s-^ 

^rtS== ill -^ g^rN 

o ^0 o 

o S 9 o"^ 5 o 3 ^o 




\ 



Fig. 501. 



u 



ROGERS' DRAWING AND DESIGN. 



343 




Fig. 502. 



344 



ROGERS' DRAWING AND DESIGN. 



The Galloway boiler will be under- 
stood from the part section shown in Fig. 
50S. Here water tubes are placed within 
the furnace tube ; holes are cut opposite 
each other in the furnace tube, and the 
joints made good by riveting the flanges 
of the water tube around the hole ; they 
pass directly across the furnace tubes, so 
that the hot gases have a considerably 
increased heating surface to act upon. 

Instead of extending through the whole 
length of the boiler, the two furnace tubes 
unite just behind the bridge wall in one 
large flue, which extends to the rear head 
of the boiler. 

A locomotive or fire box boiler of a semi- 
portable character is shown in Figs. 509 
and 510, which exhibit a half end eleva- 
tion and half cross section through the 
fire box. The rectangular fire box which 
constitutes the furnace of this boiler is 
riveted to the front part of the cylindrical 
boiler. A space called the water leg is 
left around the fire box and the boiler 
shell ; this space, which is usually from 
2^ inches to 4 inches wide, is intended 
to be filled with water. 



. ^ v^'^^^^^^^'.^^v^^<■^^■^^^^^'^^^^^^^^'^'•^s^^^^^'^^ 




ROGERS' DRAWING AND DESIGN. 



345 



Fio. 501. 



'i'^/' 



fy^. 







4 
f 

i 






1 7"^ 



///'/////f/yy///'/ / // /, / /f ^////^/, ///,//////, //\///'//'// y///y'//,//// /' / //^ y / /■ / /v\ 
'' y " L^_^_iJ yyyyy'y/''y y y /y^y' ' y ' y ' ' ' ' y ' ' ' ' y ^^^jyyjfy ^frffy/^i ^f/f^ ,/>/.y/,/ ^,,^ ///,,/\i\ 



346 



ROGERS' DRAWING AND DESIGN. 



Since the flat sides of the furnace and shell are 
liable to bulge underpressure, they must be securely 
braced or stayed ; the illustration shows the s^aj/ bolts 
which are there for that purpose. The top of the 
fire box is strengthened in a similar manner, as is 
seen in the longitudinal section of this boiler, Fig. 
510; a large number of tubes pass through the 
boiler and are secured to a tube plate at the rear end 




Fig. tOo. 

of the boiler and to the fire box at the front end ; a 
cylindrical smoke box is fastened to the rear end of 
the boiler ; the gases of combustion pass directly 
from the furnace through the tubes to the smoke 
box and thence to the smoke stack. 

Vertical boilers have the advantage of taking up 
comparatively small floor space. They are made in 
a great many varieties of designs and are used par- 
ticularly for fire engines, hoisting engines, etc., and 
wherever space is limited. 



An interesting example of this type of boiler con- 
struction is shown in Figs. 511, 512 and 513; this 
design has given excellent economical results. It 
may be observed that there are no flat surfaces 



000 


000 


000 




FiQ. 506, 



which require staying, the top of the shell, as well as 
the upper plate of the fire box, are of hemispherical 
shape, giving the maximum strength for a given 
weight of material. 



ROGERS' DRAWING AND DESIGN. 



347 



The products of combustion pass from the fire box 
through the inclined oval-shaped flue, into a com- 
bustion chamber and thence through a very large 
number of horizontal tubes to the smoke box and 
thence to the chimney. 



The water is contained in a large number of small 
lap-welded tubes, connected in various ways to each 
other, as well as to the cylindrical drum above them. 
The water fills all the tubes and a part of the drum. 
A furnace of the usual form is placed under the front 




Fig. 507. 



As may be seen in the illustration, the combustion 
chamber is lined with fire brick. 

Whenever the generation of a large quantity of 
steam in a comparatively short time is required, 
water tube boilers are now extensively employed. 
In these boilers the steam space is limited to a cylin- 
drical shell which forms only a part of the boiler. 



end of the tubes, the products of combustion circu- 
lating around the tubes and the under side of the 
drum. An illustration of this type of boiler is shown 
in Figs. 514, 515. 

In this type of boiler the drum for the steam space 
is made comparatively large, and is located parallel 
to and above the network of tubes, which are 



348 



ROGERS' DRAWING AND DESIGN. 



inclined, as well as the drum, at an angle with a hori- 
zontal plane, so as to bring the water level to about 
one-third the height of the drum in the front and 
about two-thirds of its height in the rear. The ends 
of the tubes are expanded into large water legs made 
of wrought iron, flanged and riveted to the shell, 
which is cut out for a part of its circumference to 




Fig. m&. 

receive them. The two ends of the drum are of a 
hemispherical form and are not braced as is the case 
where flat heads are used. The water legs form the 
n£tural support of the boiler. 

The boiler is entirely enclosed by a brick work 
setting ; the furnace is situated below the front end 



of the boiler and is terminated by a bridge wall ; air 
is admitted through a channel at the bottom of the 
space behind the bridge wall, and is heated in pass- 
ing through the wall. 




Fig. 509. 



The feed water is brought through a feed pipe 
leading to the front head of the drum ; within the 
main drum is suspended a mud drum below the water 



ROGERS' DRAWING AND DESIGN. 



349 



Z' 7/i * 



^ 







8>% 




Fig. 510. 



350 



ROGERS' DRAWING AND DESIGN. 



line. The feed water enters the mud drum first, 
which is submerged into the hottest part of the water ; 
in this manner the impurities of the feed water are 
largely extracted. 

Between the horizontal rows of tubes are placed 
layers of fire brick, acting as baffle plates, forcing the 
hot gases to circulate back and forth between the 
tubes and finally flow out through the chimney 
placed above the rear end of the boiler ; the upper 
part of the main drum is protected by a lining of 
fire brick. 

When several of these boilers are used together, 
forming a battery of boilers, an additional steam 
drum is usually placed at right angles and above the 
steam drums already described. 



TO DESIGN A STEAM BOILER. 

In designing a steam boiler an engineer has to 
bear in mind the following considerations: i, its 
required strength; 2, its durability; 3, its accessibility 
for inspection, cleaning and repairs ; 4, its ability to 
perform the work ; 5, the special laws of the locality 
in which the boiler is to be used, as well as the rules of 
insurance companies; and 6, the type best adapted 
to existing conditions. 



The following example may serve to show how 
the parts of a boiler may be calculated. 

Let it be required to design a 60 horse power 
horizontE I multi-tubular boiler, to carry a working 
pressure of 150 lbs. per square inch, and which will 
be capable of sustaining a test pressure of 225 lbs. 

Let tbt: length of the tubes be 15 ft. and each 
tube havi;: an internal area of 6.08 sq. in., i. e., about 
3 inches outside diameter. The heating surface 
should b<; about 37 times the grate surface. ' j 

Let the area of the grate in sq. ft equal G 

" the lieating surface in sq. ft " H 

" the area of smoke passage through the 

tubes in sq. ft " C 

" the water space in cubic feet. " W 

" the steam space in cubic feet " S 

According to the standard recommended by a 
committee of the American Society of Mechanical 
Engineers, it is customary to rate boilers by their 
horse power, considering jo potmds of water evapor- 
ated from feed water at 100" F., under pressure of 
70 lbs. by the steam gauge, is equivalent to one horse 
power ; this is equivalent to 34^ lbs. of water evap- 
orated from feed water at a temperature of 212° F., 
into steam at atmospheric pressure. 

Now, as 343^ lbs. are evaporated for i-horse 
power, for a boiler of 60 H. P. 34.5 X 60 ^ 2,070 
lbs. must be evaporated to meet the conditions. 



ROGERS' DRAWING AND DESIGN. 



351 




lb 



■© ® ©■©-ee©<5> ©■*©•©■© ©^S- 
->§; e- @%® -®-!f}-^^>®- ©{^ ©^-sfe 
■© ©•©-©-©■©-© f -©-©■©■©■©-©-©■ 
-C# ®-©-:@;o-S ©■© ©iS:-©-^!;-©^©'^- 
-S ©&©■©■ 6-^© ©-©-©■© ©oe-,,^ 
-;®! ©©-©■©-© ©■*■ ©■©-©- ©-©-©vfro 

©■©■© ®©©©0^ ©©©■©-© "K3 

© 6©®© ©a> ©-©-©-* ©■©©> ' 
©-©^■©■®© ®-^ffi-©-©-&©©- 
®©f«©-(&-^<3<g;*-©#-3>®#;- 1 



-e 

-© © ©■©© © 



-® © ffi ©-©o- 



3r 





Ftg. 512, 



Fio. 511. 



Fig. 813. 



352 



ROGERS' DRAWING AND DESIGN. 



Evaporation per i lb. of coal : i, depends on the 
quality of the coal ; 2, the rate of combustion, and 
also, 3, the construction of a boiler. The following 
table illustrates the effect of the rate of combustion, 
according to tests with a boiler, which had a ratio 
of 25 to I of the heating surface to the grate surface. 



te of combustion in 
ounds of coal per 
sq. ft. of surface. 


Evaporation 

per lb. of 

coal. 


Evaporation per 
sq. ft. of heat- 
ing surface. 


6 


10.5 


2.52 


8 


10.4 


3-33 


10 
12 

14 
16 


10. I 

9-5 

8.9 
8.2 


4.04 
4-56 
4-98 

5-25 



The evaporation per pound of coal that takes place 
(n the different types of boiler, according to Prof. 
Hutton, is as follows : 

Plain Cylindrical 5 to 8 

Vertical 5 to 10 

Water-tube 5 to 11 

Cornish 6 to 11 

Multi-tubular 8 to 12 

Locomotive Boiler 8 to 13 

The area of heating surface of each type of boiler 
is nearly always in a constant ratio to the grate 
area Below is given a table showing ike ratio be- 
tween the grate surface and the heating surface, gen- 
erally observed in the several types of boiler : 



Ratio of Grate Surface 
Style of Boiler to Heating Surface 

Plain Cylindrical 12 to 15 

Cornish 1 5 to 30 

Cylindrical Flue 20 to 25 

Cylindrical Tubular 25 to 35 

Locomotive Tubular 50 to 100 

For the boiler which we have taken as an exam- 
ple, we will assume that 12 pounds of coal can be 
burned per square foot of grate, and that one pound 
of anthracite coal will evaporate about 9 pounds of 
water at 212° F. 

As we have found, 2,070 pounds of water will have 
to be evaporated in this boiler to give us the re- 
quired 60 H. P. 

Now, as I pound of coal will evaporate 9 pounds 
of water, the coal contained in i square foot of grate 
surface, 12 pounds, will evaporate 12 x 9^108 
pounds of water. Dividing the number 2,070 by 
108 gives the reqttired area of grate. 

■ — ^ = 19.16 sq. ft. of grate surface, 
108 ^ ^ ^ 

The area of smoke passage through the tubes to 
the grate area, C : G, is according to good practice, 
made equal to i : 8 for this type of boiler. The 
number of tubes, each 3" in diameter may be found 
in the following manner : 



ROGERS' DRAWING AND DESIGN. 



353 



19.16 



X 144 == about 56 tubes. 



8 X 6.08 
Here 6.08 is the internal area of one tube. 

Suppose our boiler is designed for a steam engine 
which is to use 40 pounds of steam at a pressure of 
70 pounds per horse power per hour, and that the 
steam space shall hold enough steam to supply the 
engine 30 seconds ; the absolute pressure is nearly 
85 pounds and the specific volume of steam at this 
pressure is 5.125. The space which will be taken 
by steam required for one horse power is, 

-12 ^X 5-125 == 1.706 cubic feet. 

60 X 60 

Let us assume that this amount of space is taken 
up by the steam required for one horse power ; then 
the total space required to contain the steam for the 
60 H. P. will be 

1.706 X 60 = 102.36 cu. ft., say 103 cu. ft. 

That is, the steam space, S, should hold 103 cu. 
ft. As the water space may be taken to equal two 
times the steam space, the water space, W, will equal 
2 X S, or 2 X 103 == 206 cu. ft. 

JJoTE. — The steam space required by a given boiler depends upon 
the purpose for which the steam is to be used. Where the steam is 
under high pressure and comparatively small quantities of it are with- 
drawn at very frequent intervals, the steam space need not be so large 
as in cases where large quantities are withdrawn, even though less 
frequently. Where the boiler supplies a steam engine, it is the general 
practice to have the steam space of such dimensions that it shall contain 
sufficient steam to supply the engine for about a half a minute. 



Adding the steam space and the water space to- 
gether, we get 309 cu. ft., for the volume of both ; this 
volume would determine the capacity of the shell, 
were there no tubes passing through it and taking 
up part of the space within ; in this case, therefore, 
the space taken up by the tubes must be added to 
the above volume. 

We have already found that the boiler will con- 
tain 56 tubes. The outside area of one tube of the 
given size, is 7.107 sq. in., consequently 



56 X 7.107 X 15 
144 



41.45 cu. ft. = the space 



taken up by the tubes. Adding this to the volume 
found above, 309 cu. ft. plus 41.45 = 350.45 cu. ft., 
the entire volume of the shell. 

As the shell is to be 15 feet long, the area of its 

head must be ^^ '^^ zz.^ 3362.3 sq. in. Divid- 
ing this by 0.7854, we get the square of the diameter 

of the shell; D^= 3362.3 _ g^_ j^^ Allowing 

0.7854 
for the space occupied by the stays, etc., we may 
take D equal to 67". 

One-half of the outside surface of the shell equals 

_Z 3-14 i^ 131.4 sq. ft. The inside surfaces 

12 

of the tubes equal -^ ^ ^- — 61 1.8 sq. ft, 

12 



354 



ROGERS' DRAWING AND DESIGN. 



Allowing for the heating surface, one-half the sur- 
face of the shell, and all of the inside surfaces of 
the tubes, we have 131.4 + 6i 1.8 = 743.2 sq. ft. 

H. = 743.2 sq. ft. 

We have already found the grate surface to be 



HORSE POWER OF THE STEAM BOILER. 

As was stated before, the evaporation of 30 lbs. 
of water per hour, from a feed water temperature of 
100° F. into steam at gauge pressure of 70 lbs. is 




equal to 19.16 sq. ft. Dividing the heating surface 
by the grate surface, we are to get, according to the 
conditions given in the problem, the ratio of 2)7- 

'^^" — 38.7, or very nearly as required. 
1 9. 1 6 



the value of a commercial horse power, adopted by 
the A. S. M. E. Different boilers will generate 
steam at different pressures, receiving also the feed 
water at different temperatures. In order to com- 
pare properlj' the performances of different boilers, 



ROGERS' DRAWING AND DESIGN. 



355 



their actual evaporation must be reduced to an 
equivalent evaporation from and at 212° F. The 
problem may be stated differently as follows : It is 
necessary to find what would be the evaporation if 



Rule : From the total heat of steam at pressure 
of actual evaporatioji, subtract the observed tempera- 
ture of the feed water and add J2 ,- multiply the 
result by the actual evaporation and divide by g66 , i. 




Fig. 515. 

the feed water would be at 212° F., and the deliv- Example: If a boiler generates 2,000 lbs. of 

ered steam at o gauge pressure. steam per hour at a pressure of 100 lbs., and if the 

To find the equivalent evaporation of a boiler, j temperature of the feed water is 70°, what is the 

proceed as follows: equivalent evaporation of the boiler? 



356 



ROGERS' DRAWING AND DESIGN. 



From the steam pressure table given below, the 

total heat corresponding to a pressure of lOO lbs. 

gauge is 1184.5; consequently the equivalent evap- 

- (1184.5-70 + 32) X 2,000 

oration is ^^ ^^^^^ — - — ^\ 

966.1 

To find the horse power of the boiler, divide the 

equivalent evaporation by 34.5. In this case, the horse 



power 



(1184-5-70 + 32) X2.000 _ ^g_g ^^^^jy_ 



g66.i X 34.5 

Let W equal the actual evaporation 

H " total heat 

t " observed temperature of the feed 

water ; then, according to the above rule, the equiv- 

1 . .- • w (H— t + 32) X W 
alent evaporation is equal to ^ r^ ^ 



or 



H— t + 3: 
966.1 

The quantity 



X W 
H- 



•t + 32 



966.1 



966.1 



which changes actual 



evaporation to equivalent evaporation from and at 
212° F. is called the factor of evaporation. 

The equivalent evaporation is equal to the actual 
evaporation of the boiler, multiplied by the factor of 
evaporation; knowing the actual evaporation, and 
having a table of factors of evaporation, we are eas- 
ily able to calculate the equivalent evaporation, or 
the horse power of the boiler. 



TABLE OF GAUGE PRESSURE AND TOTAL HEAT. ' 

The following is a table of steam pressures ; the 
table gives the pressure of the steam by gauge and 
the corresponding total heat required to generate 
one pound of steam from water at 32° F. under 
constant pressure. 



Pressure by 
Gauge 




Total 
Heat 

[I46. I 


Pressure by 
Gauge 

85 


Total 
Heat\ 

[i8u4 


5 


[I5O.9 


90 


[182.4 


10 


[I54.6 


95 


ti83-5 


15 

20 ] 

25 1 


[I57.8 
[I60.5 
[162.9 


100 

105 
110 


[184.5 
[185.4 
[186.4 


30 1 

35 


[165.I 
1167.1 


115 
120 


1187.3 

1x88.2 


40 1 


[ 169.0 


125 


[ 189.0 


45 J 


170.7 


130 ] 


189.9 


50 ] 


172.3 


135 1 


190.7 


55 i 


173-8 


140 1 


[191-5 


60 1 


175-2 


145 ] 


192.2 


65 i 


176.5 


150 ] 


[192.9 


70 1 


177.9 


155 1 


193-7 


75 1 
80 1 


179. 1 
180.3 


160 ] 


194.4 



ROGERS" DRAWING AND DESIGN. 



357 



SAFETY VALVE RULES. 

The safety valve provides for the safety of boilers, 
by allowing the steam to escape when its pressure 
exceeds a certain limit. The valve is kept in its 
seat, either by a weight at the end of a lever, as in 
Fig. 516, or by a heavy weight placed directly over 



Let AB represent the length of the lever in inches. 




AC 



W 
w 
P 



Let a 
" V 



the distance between the center 
of valve and A, also in inches. 

the weight in pounds. 

the weight of the lever in pounds. 

the pressure of the steam per 
sq. in. 

the area of the valve in sq. in. 

the weight of the valve in pounds. 



Fig. 516. 

the valve, or by a strong spring. A good safety 
valve must allow all excess of steam to escape as 
fast as it may be generated. 

The valve shown in Fig. 516 rests on a circular 
seat. To find the weight, W, or the length of the 
lever, AB, for a given pressure of steam : 



If the weight of the lever and valve be 
neglected, we have, when the steam reaches the 
limit of pressure, for which the valve is intended, 

AB 

a downward pressure of W X -— and at the 

same time an upward pressure equal to P X a. 

When the valve is just about to lift, these two pres- 

AB 
sures may be considered equal ; then W X ^ =^ 

AC 



P X a; from this, W: 



P X a X ~-~- in pounds. 
AB ^ 



Taking into consideration the weight of the valve, 

which should be done for accurate practice, we have a 

AB 
downward pressure of W X ~— , the pressure due to 

AB 

the weight W, plus w X — —=;, the pressure due to the 

2 1\\^ 



358 



ROGERS' DRAWING AND DESIGN. 



weight of the lever, assuming that the weight of the 
lever acts downward in its middle, and plus Y, the 
weight of the valve. 

The upward pressure remains, as before, P X a. 

w X — ^^ + V = P X a. 



Here again, W X 



AC 



AC 



Example : Let it be required to find the weight, 
when the lever AB is equal to 36 in., AC equals 4 
in., w equals 5 lbs., V. equals 3 lbs., P equals 80 lbs. 
and a equals 6 sq. in. 

The weight of valve and lever must be taken into 
account. According to the above formula, 



Wx 3^+ 5 x-2^^ 
4 2X4- 

W X 9 + 4.5 X 5 + 3 
W — 480—25-5 



3 = 80 X 6, or 



= 480, or 

50.5 lbs. 



THE STEAM INJECTOR. 

The injector is an instrument, by the aid of which, 
the energy of. a jet of steam from the boiler, is 
utilized in forcing a stream of water into the boiler. 

The injector has largely replaced other appliances 
for feeding steam boilers, for when the work of an 



injector is compared with that of a steam pump, we 
come to the conclusion that even if the injector may 
consume a little more steam than the pump, the heat 
is returned to the boiler, by being imparted to the 
feed water. 

A boiler is tapped at the highest point of the 
steam space, and a pipe leading downward is inserted 
into the opening. To the open end of this pipe is 
attached the injector, which again is connectec^ with 
the lower part of the boiler, into which it is to force 
the feed water which it receives through a special 
pipe fr^m the source of water, be it a tank, well, etc. 

The live steam which enters the injector and is 
given the shape of a pointed jet, forms a partial 
vacuum within a chamber in the injector just above 
the feed water pipe, allowing the water to enter the 
chamber, where it acquires a velocity equal to that 
of the jet of the entering steam, and being thus 
enabled to overcome the pressure within the boiler, 
by its momentum, it is forced through an opening 
and a check valve, into the boiler. 

While the pressure within the boiler may be taken 
to be pretty nearly equal in all its parts, the partial 
vacuum caused by the condensation of the jet of 
steam meeting the colder water in the injector, com- 
pels the jet of steam to rush into the injector at a 
much higher velocity than if it were discharged into 



ROGERS' DRAWING AND DESIGN. 



359 




Fig. 517. 



360 



ROGERS' DRAWING AND DESIGN. 



the atmosphere. Consequently the high velocity 
and the resulting momentum of the entering feed 
water. 

The accompanying illustrations, Figs. 517 and 
518, show an outside view and a longitudinal section 
of an injector. Referring to the sectional view, it 
will be seen that the injector consists of a case with 
a steam inlet at its upper part, water inlet directly 
below the steam inlet, delivery outlet to the boiler 
and an overflow opening at the bottom of the injec- 
tor. Separate handles are provided to regulate the 
flow of steam, of feed water and delivery. 

The nozzles within the injector may be termed, 
according to their purpose : 

(i.) The steam nozzle, through which the steam 
enters into the chamber of the injector. It is bored 
out straight in the middle and slightly conical to- 
wards its ends. 

(2.) The combining nozzle is nearest to the 
steam nozzle ; here the steam and the feed water 
come together. This nozzle is placed in line with 
the first one. 

(3.) The condensing nozzle, is the next one ; it 
forms the vacuum, upon which is based the velocity 
of the feed water ; from this nozzle the water is 



driven into the (4) delivery nozzle, through which 
it enters the boiler. The delivery nozzle is usually 
made with the smallest bore of all the nozzles. The 
diameter of the bore in the delivery nozzle determ- 
ines the volume of water which may be forced 
through it into the boiler. The size of an injector 
is always given by the diameter of the smallest part 
of the bore in the delivery nozzle, expressed in mil- 
limeters ; thus a No. 6 injector has an opening of 6 
millimeters in diameter. 



^ 



To start the injector, open the water valve first. 
When the water appears in a solid stream in the 
overflow, open the steam valve, situated directly 
above the jet, and close the jet valve. The steam 
valve must always be opened slightly before closing 
the jet valve, so as not to break the vacuum of the 
injector. 

It will be noticed that the injector is put together 
in such a manner as to render feasible all repairs 
within by unscrewing the connected parts. The 
nozzles may have to be replaced from time to time, 
as they have to withstand the great velocity of the 
flow of water, which because of its impurities, 
occasions considerable wear on the nozzles. On 
account of this, all nozzles are made of a special 
hard metal. 



ROGERS' DRAWING AND DESIGN. 



361 




Fig. 518. 



362 



ROGERS' DRAWING AND DESIGN. 



STEAM ENGINE. 

The steam engine is a machine designed to trans- 
form the energy of steam, underpressure, into actual 
energy in the form of continuous rotation. 

For this purpose the steam is made to move the 
piston in the steam cylinder backward and forward, 
by bringing the steam into the cylinder, alternately 
from one side of the piston and then from the other, 
thus imparting a reciprocating motion to the piston. 

The mechanism which regulates the direction of 
the steam into the cylinder is called the valve 
meclianism or valve gear of the steam engine. 

When the piston, or the area which receives the 
pressure of the steam, travels in a circular path con- 
tinuously in one direction, the engine is termed a 
rotary steam engine or the steam turbine. 

The rcciprocatijjg steam engine^ in which the pis- 
ton travels back and forth, is the ordinary form of 
this important motor. It has been found to be 
the most convenient and most economical design, 
hence we shall take up for illustration and explana- 
tion this form, only, of the steam engine. 

The reciprocating motion of the piston may be 
transformed into a continuous rotary motion in 
various ways ; the crank motion is the most popular 
form of mechanism adopted for this purpose ; the 
motion of \.]\& piston is transmitted hy the piston rod, 



which is fastened firmly at one end to the piston, to 
the crosshead, from which the co7i7tecting rod leads 
to the crank pin oi the cra^ik; this forms a solid 
structure with the main shaft of the engine. 

The ?naifi shaft, receiving in this manner the 
rotary motion, serves as the source of rotary power, 
used for the many purposes of modern industry. 

The length of the cylinder is made equal to the 
travel of the piston, which itself is equal to-, twic« 
the effective length of the crank, phis the thickness 
of the piston, to which must be added the allowance 
for clearance at each end, so that the piston shall not 
strike the head of the cylinder, and at the same time 
will provide the necessary space for the steam to 
get behind the piston when the latter is at the end 
of its stroke. 

The length of the piston rod must be sufficient to 
permit the piston to return the full length of its 
stroke and still leave enough of the rod outside the 
cylinder to fasten it to the crosshead. To avoid 
leakage of steam through the hole in the cylinder 
head, through which the piston rod protrudes, 2. stiff- 
ing box is attached to the cylinder head. It is 
evident that the extreme length of the stuffing box 
must be added to the length of the piston rod. 

The crosshead is guided in its rectilinear path by 
close-fitting rods, bars or blocks, which are securely 



ROGERS' DRAWING AND DESIGN. 



363 




364 



ROGERS' DRAWING AND DESIGN. 



fastened to the engine bed, or form a part of the 
bed casting, in one piece, and which are called the 
cross/lead gtiides ; these must be set absolutely 
parallel to the axis of the cylinder. 

The steam cylinder, as well as all other parts of 
the steam engine mechanism, is fastened to a heavy 
casting called the engiiie bed^ which is rigidly held 
upon a solid masonry foundation by means of anchor 
rods, whenever the engine is of the stationary type ; 
in marine engines the bed plate is fastened to extra 
heavy frames forming part of the hull. 

When the crank is in a horizontal position, in the 
plane of the piston rod, and the crank-pin lies in a 
line drawn through the center line of the cylinder, the 
piston being at one of its extreme positions of the 
stroke, the engine is then said to be on its dead center , 
as the pressure of the steam upon the piston will not 
result in rotation of the crank. The rotatingr shaft is 
usually supplied with a heavy fly wheel, intended to 
store up the energy of rotation, and one of the func- 
tions of the fly wheel is to carry the engine mechan- 
ism past the dead centers. 

Fly wheels are of many different constructions, 
varying from a solid cast iron wheel of small diame- 
ter, to built up wheels of over 30 ft. in diameter. In 
modern practice the rim of the wheel is made wide 
enough to carry the belt which transmits the motion 



from the engine to the machinery. Such wheels are 
usually called belt wheels m distinguishing them from 
the old-style narrow rimmed fly or bnlance wheel, 
which was constructed independent of the pulley 
carrying the belt. 

In Figs. 519 and 521 are shown two views of a 
belt wheel of modern construction, 18 ft., 8 in. in 
diameter. It is unnecessary to show the entire 
wheel, as it would be simply a repetition of similar 
parts. In most cases only a quadrant of the wheel 
is shown, accompanied by a partial section like that 
shown in Fig. 519. 

It will be noticed that the outline of the belt 
wheel hub is a regular dodecagon, its sides forming 
the planed surfaces, to which the arms are attached 
by means of flanges and bolts. The other ends of 
the arms are also flanged and connected to the dif- 
ferent rim sections as shown in Fig. 520. The hub 
in this case is made in two sections and the rim in 
six sections, to each of which latter two arms are 
flanged. 

The sections of both hub and rim are held to- 
gether by bolts passing through projecting flanges, 
as shown in Fig. 520 and in partial detail in Figs. 
521, 522 and 523 . Where extra strength is re- 
quired, the reinforcements shown in Fig. 520 are 
often made use of. These consist simply of I-shaped 



ROGERS' DRAWING AND DESIGN 




366 



ROGERS' DRAWING AND DESIGN. 



pieces made of mild steel, which are shrunk into re- 
cesses of similar shape, making a rigid joint. 

A different type of reinforcement is illustrated in 
detail in Figs. 521, 522 and 523. Here, instead of 
the I-shaped piece, a wrought iron link is substituted, 
which is shrunk over bosses cast into the castings for 
this purpose. These bosses are cast so that they will 
not project beyond the surface of the rim, as shown 
in Fig. 522, leaving them flush so the wheel may 
be faced off after mounting. This method of rein- 
forcement is rapidly finding favor for permanently 
connecting parts of heavy machinery, such as large 
sectional engine beds, etc. 

It is very difficult, however, to disconnect such 
fastenings, for, when heating the link to expand it 
for removal, the lupfs will also be heated and ex- 
panded, necessitating in most cases cutting and 
destruction of the link. 

In Fig. 524 is shown a section through an arm of 
the wheel, and Fig. 526 illustrates a section through 
the rim, close to one of the flanges to which the arms 
are attached. 

Fig. 525 represents part of the hub, showing the 
face of the joint. 

The design of a fly wheel is one of the most diffi- 
cult tasks that an engineer m.ay meet and requires 
judgment and much practical experience. Often- 



times the success of an engine depends upon the 
fly wheel, for even a good and active governor would 
be unable to steady the motion of an engine with a 
variable load, were the fly wheel not able to carry 
the crank past the dead centers, as well as those 
points during the revolutions, where but little rotary 
motion is supplied by the driving parts. 

Great care must also be taken to have the material 
in the rim evenly distributed, for if the wheel is not 
balanced, the centrifugal force will have a tendency 
to bend the shaft, besides causing severe vibration. 

The rim speed of a cast iron fly wheel ■iVovX^ never 
exceed one mile, 5,280 feet per minute, for the cen- 
trifugal force has a tendency to burst the rim and 
this tendency increases as the speed increases. This 
danger is not overcome by increasing the thickness of 
the rim, for while the extra, thickness adds strength 
to the rim, the extra weight increases the centrifugal 
force. 

Two or more cylinders are often used in one steam 
engine ; when two cylinders are used, they are usually 
arranged so that the two cranks of the separate 
cylinders are at an angle of 90° to each other. When 
three cylinders are used the cranks will be at angles 
of 120° to each other. 

Such multi-cylinder engines do not require as 
heavy a fly wheel as a single cylinder engine of the 



ROGERS' DRAWING AND DESIGN. 



367 




Fig. 527. 




Fig. .528. 



same power, as the cranks will assist each 
other over the dead centers; at the same 
time they act so that one cylinder develops 
its maximum power, while the others are 
nearing completion of the stroke. 

The connecting rod of an engine is usu- 
ally made from four to six times the length 
of the crank. 

The top view of a horizontal engine is 
shown in Fig. 527. The same engine is 
shown in Fig. 528. 

One illustration shows a rio;ht hand and 
the other illustrates a left-hand engine. A 
rtght-hand engine is one in which the Jiy 
wheel is to the right of the observer as he 
stands at the head end of the cylinder looking 
towards the main bearing of the engine. 

The size of the engine is commercially 
rated by the length and diameter of the steam 
cylinder. 

When an engine produces a rotary motion 
of the fly wheel and crank, so that the crank 
when starting from its inner dead center 
rises above the axis line, or descends below 
it, upon the beginning of the stroke, the 
engine is said to "run over" or "run 
under," Fig. 529. It is, as a rule, desirable 



368 



ROGERS' DRAWING AND DESIGN. 



to have an engine run over, because the pressure 
of the connecting rod is then always downward, and 
is taken up by the guides and bedplate. 

An example of proportioning the main parts of an 
engine is given in Figs, 530 and 531. 

A vertical etigine for small power is shown in Fig. 
532 ; it shows a section through the cylinder and 




Fig. 529. 

valve chest, while Fig. 533 exhibits a section on a 
plane at right angles to that of the first section. 

The avoidance of cylinder wear, and still more 
the small floor space required by the vertical engine, 
have made its use practically universal for crowded 
power plants, steamships, etc. The support of such 



engines, however, is commonly not sufficiently rigid 
to prevent undesirable vibration of the moving 
parts. 

The section of the vertical engine shown in Fig. 
532 shows a type of a steam distributing valve, called 
the slide valve, one of the oldest and up to the present 
most reliable valve gears. 

The functions of a valve on a steam cylinder are 
primarily to admit the steam from the boiler^to one 
side of the piston, while the steam filling the other 
side of the cylinder is allowed to escape through 
the exhaust pipe, and second to stop the 
admission of steam at a certain point, for 
the purpose of producing its desired ex- 
pansion and finally to close the exhaust 
opening at such a point in the return 
stroke, that a certain volume of steam 
shall be left in the other side of the cylinder to be 
compressed behind the piston, to serve as an elastic 
cushion. 

Note. — The horizontal position of the engine is by far the most popu- 
lar for factories, power plants, etc., where there is considerable floor 
space. To have all the parts of an engine easily accessible and a solid 
support for the engine, as offered by a large bed, are the great advan- 
tages offered hy a horizontal engine ; while on the other hand, the ten- 
dency of the cylinder to wear vmequally is a disadvantage which cannot 
be denied. 



ROGERS' DRAWING AND DESIGN. 



369 



Diagram of 
HORIZONTAL ENGINES. 





Fig. ra). 



Fig. 531. 





Table of Dimensions Reference being had t( 


) above 


Diagram. 








SIZE or 

ENGINE 


A 


B 


C 


D 


E 


F 


Q 


H 


1 


J 


K 


L 


M 


N 





P 


Q 


R 


iZ>^24- 


lO-4h 


l4-8i 


16-51 


9 


15 


/; 


Z4 


7-fi 


4-Qi 


2-iOi 


10 


12 


3-4 


6i 


13 


8-0 


7i 


16 


i3^24 


l0-4h 


14-81 


16-51 


9 


15 


\i 


Z4- 


7-51 


4-2^ 


2-108 


10 


12 


3-4 


7 


15 


8-0 


81 


16 


iS^Z4 


lQ-7'z 


15-31 


18-01 


13 


15 


20 


2-4- 


8-3'z 


4-71 


d-Z'z 


II 


15 


4-0 


9 


16 


10-0 


9^ 


19 


i6-2S 


mi 


n-dl 


20-11 


15 


18 


iz 


2-9 


9-51 


5-3l\ 


3-6l 


IZ 


16 


4-10 


91 


20 


10-0 


10 


20 


18^28 


l2-5i 


17-iii 


20-3i 


15 


18 


23 


Z-9 


IO-3i 


5-m 


3-91 


13 


18 


4-10 


//b' 


24 


10-0 


Hi 


22 


20x30 


is-e'2 


19-91 


22-61 


20 


10 


15 


3-4 


12-Oi 


7-2 


4-9 


15 


20 


6-0 


// 


28 


12-0 


121 


24 


22-30 


I3-/2 


19-llz 


U-8'2 


20 


10 


27 


3-4 


I2-I(l'z 


7-9'z 


4-IOz 


15 


22 


6-0 


12 


34 


IZ-0 


I3i 


26 



370 



ROGERS' DRAWING AND DESIGN. 



It is evident that the opening or closing of the 
steam inlets or outlets of the cylinder must be care- 
fully timed to produce the required pressures in the 
different parts of the cylinder at the proper time. 

The valve is generally moved back and forth 
within the steam chest by the action of an eccentric, 
one type of which is shown in Fig. 536; this is 
securely fastened to the main shaft so as to turn 
together with it. In some rare cases, the eccentric 
is forged solid on the shaft, but the general practice 
is to fasten it by keying it on ; the eccentric is usually 
placed just outside of the bearing which holds the 
main shaft. 

This mechanism consists of two parts ; the eccen- 
tric proper or sheave, Fig. 536, and the eccentric 
strap. Figs. 534 and 535. The eccentric strap is 
made to fit in a groove in the face of the eccentric, 
or the eccentric fits in a groove in the strap. 

To the eccentric strap is attached the eccentric rod, 
which is connected to the valve rod, this finally con- 
necting to the valve. 

The eccentric is a form of crank, the difference be- 
tween them being that in the eccentric the crank pin 
is so large that it embraces the crank shaft. This 
is shown in Figs. 536 and 537. Twice the distance 
between the center of the crank shaft and the center 
of the crank pin is the length of a stroke produced by 
the crank. The same is true for the eccentric. 



The slide valve which has been used most exten- 
sively is the plain three-ported slide valve. Its form 
and action will be understood by reference to the 
illustrations shown in Figs. 538 and 539 ; here the 
slide valve is shown in its most elementary form, as 
a box open on its under side sliding over a plane 
surface on the outside of the steam cylinder. This 
surface is supplied with openings called steam ports, 
leading into the cylinder ; these ports, usually rect- 
angular in section, are indicated in the illustration 
by the letters S^ and S^. 

The third opening over which the slide valve 
moves, indicated by the letter E, is the exhaust open- 
ing, through which the steam escapes from the cylin- 
der. It will be seen that, when the valve is in its 
middle position, its two edges cover the two steam 
ports, while the hollow part of the slide valve is over 
the exhaust port. When the valve no more than 
covers the steam ports when in its middle position, 
the eccentric must be placed 90° in the advance of 
the engine crank. 

The illustration shows the piston at the left end 
of the cylinder, with the valve moving toward the 
right and about to open the steam port S^ to permit 
the steam to pass through this port into the cylinder, 
thus forcing the piston to the right. During this 
same time, the steam port Sg and the exhaust port 
E will be connected by the hollow part of the valve, 



ROGERS' DRAWING AND DESIGN. 



371 





e 




Fig. 532. 




Fig. 533. 



372 



ROGERS' DRAWING AND DESIGN. 



and the steam, which is contained to the right of 
the piston is allowed to escape into the exhaust. 
The valve will have moved to its extreme right 
position when the piston has reached the middle of 
the cylinder, and when the piston has reached its 
extreme right position, the valve will be in the mid- 
dle of its return stroke. Without this arrangement 
there would be no expansive working of the steam, 
as one end of the cylinder is left open for the admit- 
tance of live steam during the whole stroke of the 
piston, while the other end is open during the same 
time to the exhaust. To produce expansion, the 
valve must more than cover the steam ports when 
in its middle position. The amount which the valve 
projects outside of the steam ports, when the valve 
is in its middle position, is called the outside lap of 
the valve, and the amount which the valve projects 
on the inside of the steam ports in the direction of 
the exhaust ports, is called the inside lap. Fig. 540. 
The addition of laps necessitates a change in the 
angle of advance between the crank and eccentric, 
because the valve must be on the point of opening 
the steam port when the^ piston is at the beginning 
of its stroke, and therefore the valve must be away 
from its middle position, by a distance equal to the 
outside lap when the piston is at the beginning of its 
stroke. The angle between the eccentric and the 
crank must then be more than 90°. 



The correct position of the eccentric in relation 
to the crank is found by the construction illustrated 
in Fig. 541. Here AO represents the crank, while 
the circle CDE is the path of the center of the 
eccentric. On AO set off OB equal to the outside 
lap of the valve. Draw BD perpendicular to OB 
cutting the circle at D. Then OD is the position of 
the eccentric sheave, if the motion is in the direction 
of the arrow. If opposite, then OE is the right 
position of the sheave. The angle CO A i&a 90*^ 
angle, and the acute angle COD is called the angle 
of advance. 

When it is desirable to partly open the steam 
port just as the piston is beginning its stroke, the 
valve must be given a lead. The amount of such 
opening of the steam port at the beginning of the 
stroke, is called the lead of the valve. 

To produce the valve lead, the angle of advance 
of the eccentric must be increased, so as to make 
OB, in the above figure, equal to the outside lap, 
plus the lead. 

Let the circle ABCD in Fig. 542 represent the 
path of the center of the eccentric sheave. When 
the valve is in its middle position, moving toward 
the left, the eccentric will be in the position OC, 
moving in the direction indicated by the arrow. 
Let P be one position of the eccentric ; drop from 
the point P a line PM perpendicular to AB. Then 



ROGERS' DRAWING AND DESIGN. 



373 




Fios. 531 AITD 53S. 



374 



ROGERS' DRAWING AND DESIGN. 




the distance OMwill correspond to the distance the 
valve has moved from its middle position. Make 
QO equal to MO. If more positions of the eccen- 
tric are taken, and for each one a similar construc- 
tion employed, the points corresponding to the 
point Q, found above, will, when joined, produce a 
curve which will have the form of the two circles, 
AQON and OSBT. These circles are described 
on AO and OB as diameters. By means of these 
two circles the position of the valve may be readily 
found for any position of the eccentric. For instance, 
if OR is the position of the eccentric, the valve 
will be at a distance OS from its middle position. 

To find the position of the engine crank for any 
given position of the eccentric, say for the position 
OP, make the angle POL equal to the angle be- 
tween the eccentric and the crank, equal to a right 
angle plus the angle of advance; then OL will be 
the required position of the crank. 

If the distance OQ, equal to MO, be set off on 
OL instead of OP, and a similar operation gone 
through for a number of positions of the crank, it 
will be found that the different positions of the 
point Q, when joined, will form a curve just coincid- 
ing with the two circles described on OE and OF, 
Fig. 543, as diameters, the diameter EOF making 
the angle COE, equal to the angle of advance. 



ROGERS' DRAWING AND DESIGN. 



375 



From this diagram the position of the valve for any 
position of the crank may be found. If the crank is 
in position OL, Fig. 543, then the valve is a distance 
OQ from its middle position. 

We can also find any position of the crank for a 
given position of the valve. For instance, when the 



To determine whether the valves are set correctly, 
by means of diagrams taken of the steam pressure 
from each end of the cylinder and by observing and 
comparing the respective positions of the point:; of 
admission, cut-off, release and compression, the ap- 
paratus called the steam engine indicator is used. 




Fio. 538. 




valve is just on the point of opening or closing, 
it will be at a distance equal to the outside lap, 
from its central position. If an arc be described 
with the center O and a radius equal to the outside 
lap, 06, will be the position of the crank when the 
steam is admitted, and O62 its position when the 
steam is cut ofT. The position of the piston can 
easily be found from this. 



A description of the indicator and its use, together 
with a study of its diagrams, can not be classed with 
mechanical drawing, and would cause us to drift too 
far from our subject. A very interesting and thor- 
oueh treatise on the indicator is " Hawkins' Indica- 
tor Catechism," and the student who desires to take 
up the field of steam engineering, is respectfully 
referred to this work. 



376 



ROGERS' DRAWING AND DESIGN. 



Fig. 544 shows a valve diagram, as well as a 
theoretical form of an indicator diagram. The ra- 
dius of the eccentric equal to one-half the travel of 
the valve, is equal to the distance AO in the diagram. 
The outside lap equals OC, the lead equals b d, the 
angle of advance is equal to the angle EOC; the 

OuhideLaJi 
/r)6ide Lafi ■ 



i« =t 




Fig. 540. 



inside lap is equal in length to Og, the width of 
the steam port is given by MK, which is equal to 
gh. When the steam is admitted, the crank is in 
the position indicated by the line 06;. When the 
steam is cut off the position of the crank is 06,. 
When the steam is released the crank is in the posi- 



tion O63. Compression begins when the crank is in 
the position 06^; the port is completely open while 
the crank moves through the angle q o r, and it is 
completely open for exhaust while the crank moves 
through the angle tOs. 

To find the indicated horse power of a steam 
engine, fuuliiply the mean effective pressure in lbs. 




per sq. 
length 
piston 
divide 
P X 



Fig. 641. 



in. on the piston during one stroke.^ by the 

of the stroke in feet, then by the area of the 

and by the number of strokes per minute and 

the result by j^,ooo. 

LXAXN ^^. ^^^, 

— = Indicated H. P. of engine, 



33000 

P is the mean pressure ; L, 
A, the area of the piston ; 



where 

stroke ; 

ber of strokes per minute. The actual horse power 



the length of the 
and N, the num- 



ROGERS' DRAWING AND DESIGN. 



377 



may be taken as about ^ of the indicated horse 
power. 

To find the area of the piston multiply the square 
of its diameter by o.jS^-f.. 

C 




Fig. oi-Z. 



To find the mean effective pressure, divide an 
indicator diagram of the engitie into any number 
of equal parts, say lo, then measure the height of 



each part, half-way between the division lines, as 
shown by the dotted lines in Fig. 545. Add the 
length of all the dotted lines, and divide by the num- 
ber of divisions, in this case 10. 




Fig. .>t3. 



The mean effective pressure may also be found by 
m-easuring the area of the diagram, by means of a 
planimeter and dividing it by the length of the dia- 



578 



ROGERS' DRAWING AND DESIGN. 




Fig. 544. 



gram; multiply this result by the scale oj the indi- 
cator spring, and the product will be the mean effec- 
tive pressure. 

Example: Find the horse power of an engine 
when L equals 4 ft.; diameter of cylinder equals 32 
inches, P equals 40 lbs. per sq. jn. and N equals 40 
per minute. 

H. P. _4oX 4 X 804.25 X40_ 736 nearly. 
33000 

From the above formula, the proportions of a 
cylinder may be determined, when the horse power, 
pressure and nuihber of revolutions per minute are 
given. 

L X A ^ HP X 33000 j^^j.^ ^ jg ^^^ ^j.^^ ^f 
P X N 
piston, and L the length of stroke. 

The area may be expressed by the diameter of 
the piston in the following manner : A = .7854 X 
D^ ; consequently the above formula may be written 

L X .7854 X D= = ^^^^^"^ 
' ^^ P X N 

If we make L equal to d, as is often done, we have 

VTTF 



from the above formula, d = 79.59 



PN 



-from 



which we may find the required diameter of cylinder, 
for an engine which shall have a given horse power, 



ROGERS' DRAWING AND DESIGN. 



379 



mean pressure and number of strokes per minute. 
For the completed thickness of the walls of the 
steam cylinder, Prof. Reauleaux gives the following 
formula : 

Thickness ^^ Vg inch + 

lOO 

Example : If the diameter of the cylinder is 48 
in., the thickness of its wall should be 
48 



'A 



4- 



100 



= .125 + .48 = .605 



For the thickness of the heads of cylinders of 
ordinary size, when the heads are not stiffened by 
radial ribs, the thickness 

= 0.003 X d X V boiler pressure per sq. in. 

Example : If the diameter of the cylinder 
equals 40 inches, and the 1 toiler pressure 150 lb';., 
then the thickness of head ^0.003 X 40 X 1/150 
= 1.46 in. Having found a convenient thickness 
of the head and flange of the steam cylinder, upon 
which the head is to rest, the diameter of the bolts 
which fasten the cylinder head should be one-half 
the width of the flange. The number of required 
bolts may be found from the following formula : 

Number of bolts equals 0.7854 X square of diam- 
eter of cylinder, multiplied by the boiler pressure 
and divided by 5,000 times the area of a single bolt 
of the assumed diameter. 



Example : Let the diameter of the cylinder be 
equal to 32 inches ; the boiler pressure 81 lbs. per sq. 
in., and the assumed diameter of bolt equal to ^ in. 
The area of a 3/^ in. bolt equals. 0.442 sq. in., then 

Number of bolts ^-°:1 ^54 X 32 X 32 X 81 _^^^ 




Fig. 545. 



The steam chest must be made as small as the 
dimensions and travel of the valve will permit. It 
usually has the form of a square bo.x, surrounding 
the valve face. The steam chest cover, as well as 
the sides of it, are usually made of the same thick- 
ness as the cylinder walls. The size of the steam 



380 



ROGERS' DRAWING AND DESIGN. 



ports depends upon the quantity of steam which is 
to be admitted through them, and upon the speed 
of the piston. It is a good practice to make the 
area of the ports equal to yV of the piston area when 
the speed of the piston is about 600 feet per minute. 



riiickness of piston 



4 / length of stroke in in. X 
a/ diameter of cylinder in in. 
Example: If the diameter of the cylinder is 30 

in. then the required thickness of piston is 
« 4 



Q^ 



v 30 X 30 = V 900 = 5.4 in. nearly. 




When the speed of the piston is higher or lower, 
the size of the ports is increased or diminished pro- 
portionately. To find the speed of the piston, 
multiply the length of stroke by double the number 
of revolutions of crank per minute. 

A practical formula for the thickness of the piston, 
shown in Fig. 546, is : 



The piston rod may be made of wrought iron, or 
still better of steel. It is generally keyed to the 
crosshead and fastened to the piston by a strong 
thread and nut or by wedge. 

The diameter of a wrought iron piston rod may 
be found by the following rule : 

Divide the diameter of the cylinder in inches by 



ROGERS' DRAWING AND DESIGN. 



381 




Fig. 547. 



-ii- 




Fig. 548. 



382 



ROGERS' DRAWING AND DESIGN. 



60 and multiply the quotient by the square root of 
the initial steam pressure. 

Diameter of wrouoJit iron rod 
diam. of cylinder 



X V initial pressure. 




In this formula N equals the length of the con- 
necting rod divided by the length of the crank. The 
other letters represent the same values as in former 
examples in this chapter. 

Let the total area of the shoe of the crosshead 




Fig. 549. 



Fig. 550. 



Diameter of steel rod 
diam. of cylinder 

^9 



X \/ initial pressure. 



To find the pressure of the crosshead upon the 
guide, in pounds, the following may be used : 

396000 X HP 



Pressure == 



V n''— I X L X N 



be equal to A square inches, and let the pressure 
allowed upon a sq. in. be equal to p ; then, 

A pressure of crosshead upon guide 

~ P 

If the pressure of the crosshead be found to equal 
6,568 pounds, and if the pressure per sq. in. of slide 
allowed be equal to 125 lbs., then, 



384 



ROGERS' DRAWING AND DESIGN. 




Fig. 552. 



ROGERS' DRAWING AND DESIGN. 



385 




^"^ 



Fig. oo3. 



386 



ROGERS' DRAWING AND DESIGN. 




/^ 



f 



Fig. 554. 



ROGERS' DRAWING AND DESIGN. 



Bs-; 




Fig. 555. 



388 



ROGERS' DRAWING AND DESIGN. 



the area of shoe 



6568 



125 



52.5 sq. in. 



If the width of the shoe be taken equal to 4 in., 
then the length of it will be 13. i in., as 4 X 13. i 
equals 52.4 sq. in., found above (nearly). 

The pin in the crosshead which holds the connect- 
ing rod, is best made with a diameter equal to that 
of the crank pin. 

The smallest diameter of the connecting rod is 
found by dividing the diameter of the cylinder in 
inches by 55, and multiplying the quotient by the 
square root of the steam pressure per sq. in. of piston. 
The greatest diameter is one and one-half times the 
smallest. 

The diameter of the crank pin is equal to 



^ o */ HP ^ , 

'^ ^ L X N 
where 1 is the length of the crank pin journal in 
inches. 

This formula is true for a single crank and for 
one made of wrought iron. 

The length of the pin may be made equal to 
0.013 X d^ where d is the diameter of the piston. 

To find the diameter of a crank shaft for station- 
ary engines, with cylinders up to 30 in. in diameter. 



divide the diameter of the cylinder by 2 ; then sub- 
tract y^ in., and the remainder will be the diameter 
of the crank shaft. 



THE CORLISS ENGINE. 

The illustrations represent views of the cylinder 
of the Fishkill Landing Corliss engines. Fig. 551 
shows the valve gear of this engine. y^ 

A Corliss engine has four separate valves, two 
situated above the axis of the cylinder and intended 
for the admission and cut off of steam, while the 
other two are placed below the axis of the cylinder 
for the exhaust. The steam valves are rigidly con- 
nected with cranks seen on the outside of the cylin- 
der. All valves are cylindrical in form and extend 
across the cylinder above and below, respectively. 
The cranks on the outside of the valves are operated 
by a number of links, and in this manner the motion 
of the valves is actuated. 

The Corliss valve gear is used in a large number 
of steam engines. Fig. 551 shows a side elevation 
of the valve gear, while Fig. 552 exhibits a partial 
longitudinal section of the cylinder. A cross section 
being shown in Fig. 553. The cut-off mechanism 
is shown in detail in Figs. 554 and 555. In Figs. 
556, 557 and 558 the crosshead is shown. 



ROGERS' DRAWING AND DESIGN. 



389 



f- --"-" — -fi 






Figs. 556, 557 and 558. 



390 



ROGERS' DRAWING AND DESIGN. 



The disc seen in the middle of the cylinder in 
Fig. 551, called the wrist plate, is made to rock upon 
the stud in its center by a rod leading from the 
eccentric on the crank shaft. The wrist plate has 
four valve connecting rods, which connect it with 
the bell cranks, which in turn operate the steam 
valves. These valve connecting: rods can be lenorth- 
ened or shortened, so that each valve may be set 
independent of the other three. As the wrist plate 
rocks backward and forward, the exhaust valves rock 
with it. 

The two other bell cranks, which are provided 
with disengaging links, generally called hooks, are 
also given a rocking motion by the wrist plate by 
hooking in the blocks which are rigidly fastened to 
the cranks on the outer ends of the steam valve 
stems, thus causing the valves to rotate with them, 
and causing them to open the steam ports for the 
admittance of steam. 

Having turned a certain distance, the disengaging 
links on the bell crank are unhooked by a cam 
operated by the governor, and the cranks of the 
valves are pulled back to their original position by 



means of the vertical rods from the vacuum air dash 
pots. The dash pot is a cylinder in which fits a 
piston nearly air-tight. As the valve is turned it 
lifts the piston in the dash pot and creates a partial 
vacuum below it. The atmospheric pressure acts 
as a weight forcing down the piston into the dash 
pot and at the same time closing the valves. 

The air below the piston in the dash pot prevents 
a sudden shock when the piston drops down. ,As a 
consequence of this arrangement, the valves, 'are 
entirely independent in their adjustment and the 
inlet ports may be suddenly opened full width by 
the quick movement of the steam valves, while the 
exhaust valves are nearly at rest. 

The advantagfes of the Corliss valve g-ear are the 
large port area, the little friction through the valves, 
short lengths of ports, quick opening and closing 
of valves_and easy adjustment. However, the great 
number of parts makes the expense of these engines 
greater, their operation noisy, besides which it is 
impossible to run them at a high speed. Corliss 
engines do not, as a rule, run higher than 150 revo- 
lutions per minute. 



ELECTRICITY, THE DYNAMO AND MOTOR. 



Electricity is a name derived from the Greek word electron — amber. It was discovered more than 
2,000 years ago that amber when rubbed possessed the curious property of attracting light bodies. It 
was discovered afterwards that this property could be produced in jet by friction, and in A. D. 1600 or 
thereabouts, that glass, sealing-wax, etc., were also affected by rubbing, producing electricity. 

Whatever electricity is, it is impossible to say^ but for the present it is convenient to look upon it 
as a kind of invisible something which pervades all bodies. 

While the nature of electricity is a mystery, and a constant challenge to the inquirer, many things 
about it have become known — thus, it is positively assured that electricity never manifests itself except 
when there is some mechanical disturbance in ordinary matter, and every exhibition of electricity in any 
of its multitudinous ways may always be traced back to a mass of matter. 

The great forces of the world are invisible and impalpable ; we cannot grasp or handle them ; and 
though they are real enough, they have the appearance of being very unreal. Electricity and gravity 
are as subtle as they are mighty ; they elude the eye and hand of the most skillful philosopher. 

In view of this, it is well for the average man not to try to fathom, too deeply, the science of either. 
To take the machines and appliances as they are "on the market," and to acquire the skill to operate 
them, is the longest step toward the reason for doing it, and zvhy the desired resiilts follow. 

The design, manufacture and the practical applications of dynamo electric machinery is a theme 
far beyond the scope of this work, and beyond the limits of many volumes equal to it in size ; suffice it 
to say, that the subject is as inexhaustible as it /y useful to explore ; it is especially in this, as in other 
sections of the volume, that the aim of the author has been to suggest the field of work rather than to 
try to fully explain many things needed to be known. 

393 



394 



ROGERS' DRAWING AND DESIGN. 



Fig. 559. 




K 



ROGERS' DRAWING AND DESIGN. 



395 



Fia. 560. 



xn 



m. 



n rn r 
:i I I I 

J l_l L 







r^ 







ni 



m 



396 



ROGERS' DRAWING AND DESIGN. 



Electricity, it is conceded, is without weight, 
and, while electricity is, without doubt, one and the 
same thing, it is for convenience sometimes classified 
according to its motion, as — 

/. Static electricity, or electricity at j'esc. 

2. Citrrent electricity, or electricity in motion. 
J. Magnetism, or electricity iti rotation, 
f. Electricity in vibration. 
Other useful divisions are into — 
/. Positive and 

2. Negative electricity. 
And into — 

/. Static, as the opposite of 
2. Dynamic electricity. 
There are still other definitions or divisions which 
are in every-day use, such as " frictional " electricity, 
"atmospheric" electricity, "resinous" electricity, 
"vitreous" electricity, etc. 

Static electricity. — This is a term employed to 
define electricity produced by friction. It is properly 
employed in the sense of a static charge which shows 
itself by the attraction or repulsion between charged 
bodies. When static electricity is discharged, it 
causes more or less of a current, which shows itself 

Note. — Statics is that branch of mechanics which treats of the 
forces which keep bodies at rest or in equilibrium. Dynamics treats of 
bodies in motion. Hence static electricity is electricity at rest. The 
earth's great store of electricity is at rest or in equilibrium. 



by the passage of sparks or a brush discharge ; by a 
peculiar prickling sensation ; by a peculiar smell due 
to its chemical effects ; by heating the air or other 
substances in its path and sometimes in other ways. 

Current electricity. — This may be defined as 
the quantity of electricity which passes through a 
conductor in a given time — or, electricity in the act 
of being discharged, or electricity in motion. 

An electric current manifests itself by heating the 
wire or conductor, by causing a magnetic field around 
the conductor and by causing chemical charges in a 
liquid through which it may pass. 

Radiated electricity is electricity in vibration. 
Where the current oscillates or vibrates back and 
forth with extreme rapidity, it takes the form of 
waves which are similar to waves of light. 

Positive electricity. — This term expresses the 
condition of the point of an electrified body having 
the higher energy from which it flows to a lower 
level. The sign which denotes this phase of electric 
excitement is + ; all electricity is either positive 
or, — , negative. 

Negative electricity. — This is the reverse con- 
dition lo the above and is expressed by the sign or 
symbol — . These two terms are used in the same 
sense as hot and cold. 



ROGERS' DRAWING AND DESIGN. 



3f7 



Atmospheric electricity^ is the free electricity 
of the air which is almost always present in the 
atmosphere. Its exact cause is unknown. The 
phenomena of atmospheric electricity are of two 
kinds ; there are the well-known manifestations of 
thunder-storms ; and there are the phenomena of 
continual slight electrification in the air.best observed 
when the weather is fine ; ihe aurora constitutes a 
third branch of the subject. 

Dynamic electricity. — This term is used to 
define current electricity to distinguish it from static 
electricity. This is the electricity produced by the 
dynamo. 

Frictional electricity is that produced by the 
friction of one substance against another. 

Resinous electricity. — This is a term formerly 
used, in place of negative electricity. This phrase 
originated in the well-known fact that a certain 
(negative) kind of electricity was produced by rub- 
bing rosin. 

Vitreous electricity is a term, formerly used to 
describe that kind of electricity (positive) produced 
by rubbing glass. 

Magneto-electricity is electricity in the form of 
currents flowing along wires ; it is electricity derived 
from the motion of maonets — hence the name. 

o 



Voltaic electricity. — This is electricity pro- 
duced by the action of the voltaic cell or battery. 

Electricity itself is the same thing, or phase oj 
efiergy, by whatever source it is produced^ and the 
foregoing definitions are given only as a matter of 



convenience. 



ELECTRO-MOTIVE FORCE. 

The term is employed to denote that which moves 
or tends to move electricity from one place to an- 
other. For brevity it is written E. M. F.; it is the 
result of the difference of potential, and propor- 
tional to it. Just as in water pipes, a difference of 
level produces a pressure, and the pressure pro- 
duces a flow as soon as the tap is turned on, so dif- 
ference of potential produces electro-motive force, 
and electro-motive force sets up a current as soon as a 
circuit is completed for the electricity to floiv through. 
Electro-motive force, therefore, may often be con- 
veniently expressed as a difference of potential, and 
vice versa ; but the reader must not forget the dis 
tinction. 

In ordinary acceptance among engineers and prac- 
tical workine electricians, electro-motive force is 
thought of as pressure, and it is measured in units 
called volts. The usual standard for testing and 



398 



ROGERS' DRAWING AND DESIGN. 



comparison is a special form of voltaic cell called 
the Clark cell. This is made with great care and 
composed of pure chemicals. 

The term positive expresses the condition of the 
point having the higher electric energy or pressure, 
and, negative, the lower relative condition of the 
other point, the current is forced through the 
circuit by the (E. M. F.) electric pressure at the 
generator, just as a current of steam is impelled 
through pipes by the generating pressure at the 
steam boiler. 

Care must be taken not to confuse electro-motive 
force with electric force or electric energy ; when 
matter is moved by a magnet, we speak rightly of 
magnetic force ; when electricity moves matter, we 
may speak of electric force. But, E. M. F. is quite 
a different thing, not " force " at all, for it acts not 
on matter but on electricity, and tends to move it. 



THE DYNAMO, OR GENERATOR. 

The word dynamo, meaning power, is one trans- 
ferred from the Greek to the English language, 
hence the primary meaning of the term signifying 
the electric generator is, the electric power machine. 

The word generator is derived from a word mean- 
ing birth-giving, hence also the dynamo is the ma- 
chine generating or giving birth to electricity. 



Again, the dynamo is a machine driven by power, 
generally steam or water power, and converting the 
mechanical energy expended in driving it, into elec- 
trical energy of the ciir rent form. 

Dynamos are classified as — 
/. Unipolar dynamos. 

2. Bi-polar (or 2-pole) dynamos. 
J. Multipolar dynamos. 

This division is caused by their different construc- 
tion, but, whatever their shape or size or peculiarity 
of application, the principles upon which they work 
are always the same — a dynamo is always a machine 
for generating electric currents. 

It should be understood that an electric dynamo 
or battery does not generate electricity, for if it were 
only the quantity of electricity that is desired, there 
would be no use for machines, as the earth may be 
regarded as a vast reservoir of electricity, of infinite 
quantity. But electricity in quantity without pres- 
sure is useless, as in the case of air or water, we can 
get no power without pressure. 

As much air or water must flow into the pump or 
blower at one end, as flows out at the other. So it 
is with the dynamo ; for proof that the current is 
not generated in the machine, we can measure the 
current flowing out through one wire, and on through 
the other — it will be found to be precisely the same. 



ROGERS- DRAWING AND DESIGN. 



399 



As in mechanics a pressure is needed to produce 
a current of air, so in electrical phenomena an elec- 
tro-motive force is necesary to produce a current of 
electricity. A current in either case can not exist 
without a pressure to produce it. 

To summarize, the dynamo-electric generator or 
the dynamo-electric machine, proper, consists of five 
principal parts, viz.: 

1. The armature or revolving portion. 

2. The field magnets, which produce the magnetic 
field in which the armature turns. 

3. The pole-pieces. 

4. The commutator or colle( tor. 

5. The collecting brushes that rest on the commu- 
tator cylinder and take off the current of electricity 
generated by the machine. 

In brief, the purpose of the dynamo is to change 
mechamcal motion, applied to the armature, revolving 
it at high speed, into electrical energy. 



THE ELECTRIC MOTOR. 

An electric motor is a machine for converting 
electrical energy into mechanical energy; in other 
words it produces mechanical poiver when supplied 
with an electric current ; a certain amount of energy 
must be expended in driving it ; the intake of the 



machine is the term used in defining the energy ex- 
pended in driving it ; the amount of power it deliv- 
ers to the machinery is denominated its outpiit. 

The difference between the output and the intake 
is the real efficiency o{ the machine ; it is well known 
that the total efficiency of an electric distribjition 
system, which may include several machines, usually 
ranges from 75 to 80 per cent, at full load, and 
should not under ordinary circumstances fall off 
more than — say 5 per cent, at one-third to half load ; 
the efficiency of motors varies with their size, while 
a one-horse power motor will, perhaps, have an effi- 
ciency ot 60 per cent, a loo-horse power may easily 
have an efficiency of 90 per cent. 

A dynamo, as ordinarily constructed, consists of 
two parts ; the stationary magnet frame and the re 
volving part, or the armature. In Figs. 559 and 
560 is illustrated a multipolar generator, that is a 
dynamo having more than two poles, in this case four. 

The armature is generally the revolving part and 
to it are secured in various manners numerous loops 
of wire. The space occupied at the end of the mag- 
net poles by the armature, is called the magnetic 
field. The armature revolving in this magnetic field 
slightly charged by the influence of the magnetism 
naturally retained in the iron of the magnet frame, 
drives its numerous conductors through the magnetic 



400 



ROGERS' DRAWING AND DESIGN. 



lines, and as each of these conductors, called loops, 
passes through this magnetic field, it gathers a little 
amount of electrical energy, or as it is generally' ex- 
plained, a slight electric current is made to flow 
through it and on through the commutators and 
brushes to wires outside of the dynamo ready for 
service. 

The larger the number' of loops the greater the 
electrical energy gathered by the armature. The 
amount of the electrical energy gathered by the 
armature conductors is greatest when the magnetism 
in the magnet is highest. 

When the armature is first set in motion the elec- 
tric current in it is very mild as the magnetic influ- 
ence is also very slight at that time, but as soon as 
a mild current is produced, it is made to pass through 
insulated wires wound around the magnet cores, 
which at once strengthens the magnetism and thus 
in turn calls forth a greater electrical energy within 
the armature, which greater energy is again utilized 
to strengthen the influence of the magnet. In this 

Note. — There are a great variety of armatures in use ; the drutn 
armature has'beeu found the most popular on account of its simplicity, 
and comparative efficiency. 

One of the types is the ring armature^ whose efficiency is so low, 
that it never found very extensive favor, although it is very simple and 
easy to repair. 

For large machines, the multipolar armature^ is used almost exclu- 
sively ; still another type, disc armature. 



manner by increasing the magnetic influence, the 
electrical energy within the armature conductors is 
increased up to the desired limit. 

When electrical energy is supplied to a dynamo 
the armature will turn with great velocity and force, 
and thus the machine will transform electrical energy 
into mechanical motion. In this case the machine 
is called a motor. 

The armature in the dynamo shown Fig. 559'is of 
the drum, type ; it consists of the core, upon Which 
wires are wound, these wires being connected to the 
commutator, upon this commutator, the brushes are 
riding, which gather up the current as it is delivered 
to the commutator by the armature, whence it is led 
outside of the machine to the circuit. 

The armature core is made of sheet iron discs 
usually about 0.002 inch in thickness. The outer 
discs being unsupported at the' outer edge, are usually 
made of sheets of tV thickness. Three of these at 
each end of the core will be enough to hold the rest 
of the discs from spreading. 

These discs are usually made of the best charcoal 
iron. Between the discs a thin sheet of paper is 
laid. The circumference of these. discs is provided 
with apertures of various forms, for holding the ar- 
mature coils in place of which several types are 
shown in Figs. 561-564. 



ROGERS' DRAWING AND DESIGN. 



401 




Fig. 561. 




Fio. 562. 



Fig. 561 shows a disc with round holes to contain 
the conductors. Fig. 562 shows plain slots with 
parallel slides. Fig. 563 is a slight modification of 
the preceding form. In the slots shown in Fig. 564 
the conductors are held securely by means of hard- 
wood strips, which are driven in above them. The 
disc is punched out of sheet iron, a hole for the shaft 
and key-way being also cut out. The disc is then 
placed under a punching press with a revolving table 
for the disc, which is automatically moved a certain 
distance between each stroke of the press. Thus 
all slots are punched. 




Fig. .563. 




Fig. 564, 



Figs. 565 and 566 show a section of the commutator 
as well as a partial end-view. It consists of the shell, 
that is the outside casting, placed directly upon the 
shaft. One end of the shell is provided with a cir- 
cular projecting lip wedge-shaped in section, to 
support the segments, while the other end is provided 
with a thread. A ring, also wedge-shaped in section 
is placed on the shell near its end, for the purpose 
of supporting the other end of the segment, and a 
nut is then screwed upon the threaded end of the 
shell, pressing the ring toward the segments, and 
holding them securely. These segments are insu- 



402 



ROGERS' DRAWING AND DESIGN. 



lated from each other, as well as from the shell, by 
strips of mica or fibre, indicated by the heavy black 
lines in the illustration. The segments are made 
of hard copper. The prolongation of the segment 
at the back is called the ear or lug of the segment. 







The purpose of the lug is to provide a means to se- 
cure the conductor to the commutator and for this 
slots are cut in the lugs, just wide enough to accom- 
modate the wires, which may be secured to it by 
small screws or by soldering. The shell, ring and 



Fig. 566. 

nut are made of bronze. Clearance 
is allowed between the shell and the 
segments to secure better insulation, 
t is not necessary that the shell 
should bear on the shaft throughout 
its entire length, and to save boring, 
it is cored out, as shown in the illus- 
tration, leaving about an inch at each 
end of the shell for the support. 



ROGERS' DRAWING AND DESIGN. 



403 




Fig. fier 



Fig. 508. 



Fig. 5«9. 



404 



ROGERS' DRAWING AND DESIGN. 





FlQ. 571. 



Fig. 570. 



The commutator shell is secured to the shaft 
by means of a small key not shown in the 
illustration. 

In Figs. 567 and 568 is illustrated a section 
of a simple self-oiling bearing for the support 
of the shaft of the dynamo, illustrated in Figs. 
559 and 560. Within the bearing box is con- 
tained the cylindrical brass bushing. The 
upper part of the bushing is provided with a 
slot, and here is introduced the oil ring, which 

rests upon the shaft, dipping into the oil contained in the reservoir below. 
When the shaft is revolved, the ring takes up oil and carries it to the 
shaft. Larger bearings are provided with two oil rings. Provision 
must be made to drain off the oil and to furnish a fresh supply. 

The upper part of the bearing box is often made with a large opening 
covered by a hinged lid for the purpose of inspection, as well as for 
supply of oil. A convenient addition is an oil gauge, which shows the 
amount of oil in the reservoir. 



ROGERS' DRAWING AND DESIGN. 



405 



Fig. 573. 




Fig. 573. 



In the same illustration it can be seen that the rear 
end of the bearing box is turned to receive the brush 
frame, Fig. 569, which in this case is provided with 
four holes to receive four brushes. Another kind of 
brush frame, called a rocker arm, is shown in Fig. 5 70, 
made for two brushes. The holes for receiving the 
brush holder studs are often made square. A brush 
holder stud is shown in Fig. 571 ; it is made of brass 
and is circular in section. The section shown in 
black represents the insulating washers and insu- 
latinQ- bushinofs, made of hard rubber, insulating the 
brush holder stud from the rocker arm. Outside 
on the stud is placed a brass washer and a 
cable luof, which is used to connect it with 
the main cable or leads carrying the cur- 
rent to the point of distribution. 

The cable lug is shown separately in 
Figs. 572 and 573. A form of brush 
holder which is rapidly becoming most 
popular, is shown in Fig. 574 ; it is called 
the Reaction Brush Holder. Here the 
brush is wedded in between the brush 
holder and the commutator without any 
support on the outer side, the pressure of 
the curved lever forcing the carbon brush 
against the inclined face of the holder as well as 
against the commutator. The pressure of the lever 



406 



ROGERS' DRAWING AND DESIGN. 



is caused by a helical spring, terminating in a 
straight projection, which can be set into any one of 




Fig. 574. 



the notches on the lever, thus regulating the pressure 
of the lever to any desired degree. 



The magnet frame may be cast in one or more 
parts, together with the pole pieces, or the pole 
pieces may be bolted to the frame. The magnet 
frame must be rigidly secured to the base. The 
bearings or pedestals may be cast in one piece with 
the base, or fastened to it by bolts. The magnets 
may be made of cast iron, wrought iron, or mallea- 
ble iron, according to the requirements. 

Two methods of exciting the field are shown in 
the diagrams in Figs. 575-576. The shunt method 
of excitation consists of forming a separate circuit of 
the magnetizing coils which are connected directly 
between the brushes, or in shunt, to the external 
circuit. 

The diagram in Fig. 575 shows the manner in 
which shunt winding is accomplished. 

Another method for excitation of the field is the 
series winding. Here the entire current flowing 
through the armature is made to flow through the 
magnetizing coils. 

A combination of series and shunt winding, gives 
the compound winding, shown in the diagram, Fig. 
576. This winding is very extensively used for 
generators, but is seldom used for motors, as either 
a series or a shunt winding serves for almost all con- 
ditions of operation. 



ROGERS' DRAWING AND DESIGN. 



407 





Fk;. .'.7.-,. 



Fig. 576. 



INSTRUMENTS AND METHODS OF USE 



Preceding the section of " Lettering" and beginning at page 41 much valuable matter relating to 
the "Drawing board, T square and triangles" may be found, with many illustrations; what follows 
properly belongs with the above section, but is removed to a less important part of the volume because 
the matter is almost too elementary ; it is inserted here "lest we forget." 

Good tools are necessary for a proper output of good work but it is not always the man who has 
the most or the better tools that does the best ; a little observation also shows that every regular draughts- 
man has his own select tools, gathered as he has progressed in study and practice to suit his own "handy" 
method of work ; the time comes when the draughtsman declines the employment of any but the regu- 
lar instruments, relying upon his manual dexterity to execute all necessary drawings. 

There is an old adage to the effect that " an ounce of showing is worth a pound of telling " ; the 
kindly assistance of an experienced draughtsman at the beginning of one's efforts is invaluable and worth 
the fee that migrht be charcred. Euoene C. Peck, M. E., has written an account of the method he 
employed in teaching a class of the employees of the Cleveland Twist Drill Co.; it is quoted almost in 
full in the note below : 

Note. — The method employed was mapped out more with a view to teaching the employees to read drawings than to make draughtsmen 
of them, but at the same time so that those who cared to follow the profession in the future would be able to use all the information and prac- 
tice to good advantage ; no originality in plan of teaching was attempted ; the class consisted of twenty pupils who had been through fractions 
and percentage in arithmetic; some had taken lessons previouslv in drawing, knew the use of different instruments and understood the ordi- 
nary geometrical problems occurring in drawing, while others were without any such previously acquired knowledge. 

As very little drawing could be done in one evening in a class they were instructed to do all drawing at home. Each pupil was furnished 
with a blueprint of instructions such as would be needed outside of class, and also a plate (blueprint) to copy from. These plates were drawn, 
then blueprinted, but to a scale of about lo inches to the foot, so that no copying by dividers could be done. The first four contain the ordinary 
geometrical problems, the next four projection, cylindrical and conical intersections and developments ; then came the simple machine parts 
to teach the correct placing of views, shading, etc. From this on the plates gradually get more intricate and complicated, but in all cases are 
taken from our own shop drawings or a machine in the factory, and more especially is a drawing of a jig or fixture used which may have given 
any trouble to the machinist to read. These drawings are then made at home and left in the drawing office, where they are corrected and 
marked, a record of the progress of the student being kept for reference. 

Later they were given a little algebra in the shape of simple formulas which, by the way, gave most of them some trouble until they got 
to handle the characters as though they had no value, or to treat them by the rules regardless of their value. 

A short course in the practical laying out and working of gear problems came next, which gave very little trouble, as most of the students 
were more or less familiar with the subject. 

This year's course in class closed with logarithms, and considering that I left the theory of exponents out of the question, taught them 
only the use of the tables and gave them rules to solve the different examples by, they handled the subject remarkably well. 

413 



41J: 



ROGERS' DRAWING AND DESiGN. 



COMPASSES. 

Compasses are instruments for describing cir- 
cles, measuring figures, etc.; Figs. 577—580 show 
a pair of compasses, a pencil, a lengthening bar 
and a pen point, either of which may be inserted 
into a socket in one leg of the instrument when a 
circle in pencil or in ink is to be drawn. The other 
leg is fitted with a needle point and acts as the cen- 
ter about which the circle is to be described. 

The compasses shown in Fig. 577 have a single 
socket only; the leg with the needle-like point is called 
a divider point ; the other leg has a stationary needle 
point which is placed in the center of the circle to 
be drawn : it will be noticed that one leg of the 
compasses is jointed; this is done, so xhsiX. the com- 
pass points may be kept perpendicular to the paper 
when drawing circles. 

Note. — The student should learn to open and close the compasses 
with one hand ; those provided with a cylindrical handle at the head 
are to be held gently between the thumb and the forefinger and those 
minus the handle should be held with the needle point leg resting 
between the thumb and fourth finger, and the other leg between the 
middle and forefinger. Only one hand should be used in locating the 
needle point at a point on the drawing about which the circle is to be 
drawn, unless the left hand merely serves to steady the needle point. 
Having placed the needle point at the desired point, and with it still 
resting on it, the pen or pencil may be moved in or out to any desired 
radius. When the lengthen ng bar is used both hands must be em- 
ployed. 





Fig. 578. 



Fig. 579. 



Fr;. ,577. 



Fig. 580. 

The joint at the head of the compasses 
should hold the legs firmly in any desired 
position and at the same time should per- 
mit their being opened and closed with one 
hand ; the joint may be tightened or 
loosened by means of a screwdriver or 
spanner, which is furnished with the instru- 



NoTE. — The needle point itself, in all good instruments, is a separ- 
ate piece of round steel wire, placed in a socket provided at the end of 
the leg. The wire, as a rule, has a shoulder at its lower end, below 
which a fine, needle-like point projects. 



ROGERS' DRAWING AND DESIGN. 



415 



ment ; compasses should not be used for 
circles of too large a radius, not allowing 
the points to be placed at a right angle with 
the paper. A lengthening bar. Fig. 580, is 
used to extend the leg carrying the pen or 
pencil points, as the case may be, when 
circles of a large radius are to be de- 
scribed. 

Circles should be drawn with a continuous 
motion, with an even, slight pressure on the 
pen ; when inking in a circle it is well to 
stop exactly at the end of a single revolu- 
tion, as the line may become uneven when 
going over it a second time ; when closed, 
the needle point and the pen or pencil point 
are to be set in such a manner as to be 
even. 

DIVIDERS. 

Dividers are used for laying off distances 
upon a drawing, or for dividing straight lines 
or circles into parts ; an instrument of this 
kind is shown in Fig. 581 ; the points should 
be thin and sharp, so that they will not puncture 
holes in the paper larger than is absolutely neces- 
sary ; when using the dividers to space a line or 
circle into a number of equal parts, they should 
be held at the top between the thumb and fore- 



no. 581. 



finger, as when using compasses ; to mark off the 
spaces, the instrument should be turned alternately 
to the right and left. 

The divider shown in Fig. 581 is provided with 
a hair spring attachment, which enables the user to 
make quite ^ne adjustments ; in this case one leg is 
made separate from the main body of the instru- 
ment with its upper end terminating in a spring, 
which tends to bring one leg toward the other. 

The legs of dividers, as well as those of compasses, 
are made triangular in section, except near the 
point, where the corners are ground off sufficiently 
to make a round point. 

If the point should be left triangular, the holes 
punctured into the paper, would be bored out to 
such an extent, while turning the instrument, that 
accurate measurements would be impossible. 

It being essential that the points of dividers be 
kept in good condition, they should never be used 
for anything else, except the purpose they are made 
for. The joint ai the head of the dividers should 
be kept not too tight, for unless there is a hair- 
spring attachment, as described above, it will be dif- 
ficult to adjust the dividers accurately, owing to the 
spring in the legs. Lost motion in the head joint 
is also a very objectionable feature, and should be 
attended to as soon as detected. 



416 



ROGERS' DRAWING AND DESIGN. 



BOW PENCIL AND BOW PEN. 

A bow pencil a,ndi aubowpen are shown in Figs.583 and 
584; these instruments are made for describing small 




Fig. 582. Fio. 583. Fig. 584. 

circles. The two points should be adjusted evenly, 
that is they should be of the same length, otherwise, 
very small circles cannot be described. To open or 
close either of the above mentioned instruments; 



Fig. 585. Fig. i 



support it in a vertical position by 
resting the needle point on the paper, 
and pressing slightly at the top with 
the forefinger of one hand, and turn 
the adjusting screw or nut with the 
thumb of the same hand. Bow di- 
viders for measuring very small dis- 
tances are also largely in use, see 
Fig. 582. 

DRAWING PENS. , 

For drawing ink lines other than 
arcs and circles, the drawing pen or 
ruling pen, is used, Figs. 585 and 586; 
these consist of two thin steel blades, 
attached to a handle made of wood, 
ivory or light metal ; the points are 
made of two steel blades which open 
and close, as required for thickness of 
lines, by a regulating thumb-screw. 

When using the ruling pen it 
should be held as nearly perpen- 
dicular as possible, the hand bearing 
slightly on the tee square or the tri- 
angle against which the line is drawn. 
The pen m-ust not be pressed against 
the edge of the tee square or triangle 



ROGERS' DRAWING AND DESIGN. 



417 



as the blades will then close together, thus making 
the line uneven. The edge should only serve as a 
guide ; the pen should be held with the thumb- 
-screw on the outside. 

BEAM COMPASSES. 

For describing very large circles beam compasses 
are used ; these compasses are shown in Fig. 587, 
with a portion of the wooden rod or beam on which 
they are used. 

At A, Fig. 587, is shown a section of the beam, 
which has the shape of a letter T. This form has 
considerable strength and rigidity. Beam com- 
passes, as shown in Fig. 587, are provided with extra 
points for pencil or ink work. While the generil 
a Ijustment is effected by means of the clamp against 
the wood, minute variations are made by the screw 
B, shifting one of the points. 

Note. — A good drawing pen should be made of properly tempered 
steel, neither too soft nor hardened to brittleness The nibs shoxUd be 
accurately set, both of the same length, and both equally firm when in 
contact with the drawing paper. The points .should be so shaped that 
they are fine enough to admit absolute control of the contact of the pen 
in starting and ending lines, but otherwise as broad and rounded as 
possible, in order to hold a convenient quantity of ink without drop- 
ping it. The lower (under) blade should be suificienth- firm to prevent 
the closinp^ of the blades of the pen, when using the pen against a 
straight-edge. The spring of the pen which separates the two blades 
should be strong enough to hold the upper blade in position, but not so 
strong that it will interfere with easy adjustment of the thumb-screw ; 
the thread of the thumb-screw should be deeply and evenly cut so as not 
to strip. 



This instrument is quite delicate, and, when in 
good order, is very accurate. It should be used only 
for fine work on paper, and never for scribing in 
metal. 




Flu. 387. 



DRAWING INK. 

Liquid India ink can be procured in bottles with 
glass tube feeders, as in Figs. 588 and 589, or with a 
quill attached to the cork, by means of which the pen 
may be filled by drawing it through the blades; a 
common writing pen may a'so be used for filling the 
pen in the same manner as described for the glass 
feeder or quill. 

Dry ink of good quality however in sticks, Figs. 
590-593, cannot be surpassed, although it requires 



418 



ROGERS' DRAWING AND DESIGN. 



skill in its preparation. In case the stick ink is used 
put enough clean, filtered or distilled water in a shal- 
low dish or "tile" for making enough ink for the 
drawing in hand ; place one end of the stick in 
the water, and grind by giving the stick a circular 
motion. Do not bear hard upon the stick. Test the 
ink occasionally to see whether it is black. Draw 
a fine line with a pen and hold the paper in a strong 
light. If it shows brown or gray grind a while 
longer and test again. Keep grinding until a fine 




Fig. ] 



line shows black ; the time required to obtain the 
desired result depends entirely on the amount of 
water used. The ink should be kept free of dust 
and prevented from evaporating by covering it with 
a flat plate of some kind. 




Fig. 589. 




Figs. 590-593. 

Note. — If stick ink is used it is very good policy to buy a stick of 
the very best quality, costing, say about a dollar, as, perhaps, it will 
last longer than several dollars' worth of liquid ink. The only reason 
for using liquid ink is that all lines are then sure to be of the same 
blackness, and time is saved in grinding. 



ROGERS' DRAWING AND DESIGN. 



419 



When trouble is caused by the ink drying between 
the blades and refusing to flow, especially when 
drawing fine lines, the only remedy is to wipe out 
the pen with a cloth. Do not lay the pen down for 
any great length of time when it contains ink ; wipe 
it out first. The ink may sometimes be started by 
moistening the end of the finger and touching it to 



RULES AND SCALES. 

The rule is used for measuring and comparing 
dimensions; they are divided in inches, halves, quar- 
ters, eighths, sixteenths, and thirty-seconds. 

For some purposes the rules as explained above 
cannot be used, as i. e., for making drawings smaller 





1 


II II Tip 

2963 


° 6 


' ^ 5 


3 4- 

.4 


3 6 

3 


7 8 

2 


9 10 


1 


11 12 




llllll 
3 s 9 


^ 




3/jlNCH 

HON 


C 




?<^INCH 




v_ 


2 


t It 01 6 

ill ill ill III III ill 


8 /. 9 

llllllllllll llllll 


III llllll lllllllll 


3 L 

ill llllllllllll ill! 







t 9 

iliilillillilllili 


6 

lllllllll 


lllllllll! 


I. 



Fig. 594. 



the point of the pen, or by drawing a slip of paper 
between the ends of the blades. 

Before using the pen it is well to try it first on a 
piece of paper to make sure that it will produce 
lines of the required thickness ; the border of the 
sheet of paper on the drawing board may be used 
for this purpose, according to long established 
custom. 



or larger than the actual size of the object to be 
drawn. Scales are then employed as shown in Figs. 
594 and 595. 




\ mv^ 



Fig. 595. 



420 



ROGERS' DRAWING AND DESIGN. 



The most convenient forms are the usual flat or 
triangular boxwood scales, having beveled edges, 
each of which is graduated for a distance of twelve 
(12) inches. These beveled edges serve to bring 
the lines of division close to the paper when the 
scale is flat, so that the drawing may be accurately 
measured, or distances laid off correctly. 

A very convenient form of scales is shown in Fig. 
595. It represents a triangztlar 
scale (broken). It reads on its 
different edges as follows : 

O 

(a) 3 inches and 13^ inches to 
one foot, I inch and 3^ 
inch to one foot, (b) Y^ 
inch and 2/^ inch to one 
foot, i^ inch and y% inch 
to one foot and (c) one 
edge reads sixteenths 
the whole 12 inches of 
its length. 

PROTRACTOR. 

A protractor x?, shown in Fig. 596 ; it is an instru- 
ment for laying off or measuring angles on paper, 
or for dividing a circle into an equal number of 
parts ; it is also used in connection with a scale to 
define the inclination of one line to another. 




The outer edge of the protractor is a semicircle, 
with center at O and, for convenience, is divided 
into 180 equal parts or degrees from A to B and 
from B to A. Protractors are often made of metal, 
in which case the central part is cut away to allow 
the drawing under it to be seen. 

When using the protractor it must be placed so 
that the line O B, Fig. 596, will coincide with the 
line forming one side of the an- 
gle to be laid off or measured, 
and the center O must be at the 
vertex of the angle. 

IRREGULAR CURVES 
AND SWEEPS. 

Curves are irregular 
lines ; a circle is a resfu- 
lar line. Curves other 
than arcs of circles are 
drawn with the pencil or 
ruling pen by means of 
curved or irregular-shaped rulers, called irregular 
curves or szueeps, Figs. 597-608. These are made of 
various materials, wood, hard rubber or celluloid, in a 
great variety of shapes. A certain number of points 

Note. — A whole circle contains 360 degrees, a right angle contains 
90 degrees and therefore as many as a. % ol a circle. A 45 degree angle 
contains as many degrees as j^ of a circle. 



ROGERS- DRAWING AND DESIGN. 



421 




Figs. 597 to BOB. 



422 



ROGERS' DRAWING AND DESIGN. 



is first determined througii which the line is to pass, 
and said Hne should be first sketched in lightly, 
freehand. The irregular curve is then applied to 
the curved line so as to embrace as many points as 
possible ; only the central points of those thus em- 
braced should be inked in; this process is continued 
until the desired curve is completed. 

It is very difficult to draw a smooth 
continuous curve. In order to avoid 
making the line curve out too much 
between the points or to cause it to 
change its direction abruptly where 
the different points join, the irregular 
curve should be fitted so as to pass 
through three points at least, and, 
when moving it to a new position, by 
setting it so that it will coincide with 
part of the line already drawn. When 
neatly penciled over, after having 
been sketched in free-hand, little 
difficulty will be experienced to ink 
it, the pencil line showing the direction in which 
the curve is to be drawn. 

When inking with the irregular curve, the blades 
of the pen should be kept against it and the thumb- 
screw on the outside ; the inside flat surface of the 
blades must have the same direction as the curve at 




Fig. 609. 



Fig. 610. 



the point where the pen touches the paper. It will 
be readily understood therefore that the direction of 
the pen must be continually changed, 

PENCILS. 

Drawings are generally made in pencil and then 
inked. A hard pe^icil'x?, best for mechanical drawing. 
The pencil should be sharpened as shoWn in Figs. 609 
and 610. Cut the wood away, about Y^ or 3/^ of an 
inch of the lead projecting ; then sharpen it flat by 
rubbing it against a fine file or apiece of fine emery 
cloth or sandpaper that has been fastened to a flat 
stick. Grind it wedge-shaped as shown in the figure. 
If sharpened to a round point, the point will wear 
off quickly and make broad lines, thus making it 
very difficult to draw a line exactly through a point. 
The pencil for the compasses should be sharpened 
in the same manner but should have a narrower 
width. 

The pencil line should be made as light as pos- 
sible ; pressing the pencil too hard will often cut the 
paper or leave a deep mark which cannot be erased. 
The presence of too much lead on the surface of the 
paper tends to prevent the ink passing to the paper 
and in rubbing out pencil lines the ink is reduced 
in blackness and the surface of the paper is rough- 



ROGERS' DRAWING AND DESIGN. 



423 



ened, which is a disadvantage. As little erasing or 
rubbing out as possible should be done. 

Lines are drawn with the flat side of the lead 
pressed lightly against the straight-edge, as close to 
it as possible, the pencil being held almost vertically. 

DRAWING PAPER. 

The first thine to be considered in select'mir draza- 
ing paper is the kind most suitable for the proposed 
plan. The qualities that constitute good paper are 
strength, uniformity of thickness and surface, neither 
repelling nor absorbing liquids, admitting of consid- 
erable erasing without destroying the surfaces, not 
becoming brittle nor discolored by reasonable ex- 
posure or age, and not buckling when stretched or 
when ink or color is applied. 

The sizes and names of commercial drawing paper 
made in sheets is as follows : 

Cap 13x17 ins. 

Demy 1 5x20 

Medium 17x22 

Royal 1 9x24 

Super Royal 19x27 

Imperial 22x30 

Atlas 26x34 

Double Elephant 27x40 

Antiquarian 30x53 



For large drawings paper is made in rolls. " Detail 
paper" is especially made for marking out new 
designs ; it is made in rolls 36, 42, 44, 48 and 54 
inches wide ; the size of detail drawings for shop 
use, of course, are dependent upon the type of the 
drawing, the size of the parts detailed and the scale 
to which they are drawn ; the following sizes are 
good average ones as they can be cut very economic- 
ally from the rolls sold in print shops : 6 x 9, 9 x 1 2, 
12 X 18, i8x 24, 24 X 36, 36 X 48 and 48 x 72 inches. 

PREPARING FOR WORK. 

The paper is first secured to the drawing board 
by means of thumb tacks, one at each corner of the 
sheet. It should be stretched flat and smooth ; to 
obtain this result proceed as follows : press a thumb- 
tack through one of the corners about y^ inch or ^ 
inch from the edge. Place the tee square in position 

Note. — Border lines such as are used throughout the pages of this 
book are frequently of considerable service to the draughtsman but 
they must be used with a sense of "the fitness of things." Thus: 
border lines are out of place in working drawings, etc., but where a set 
of drawings are to be inspected and important contracts decided upon 
by non-technical business men or capitalists, a neat border line is often 
the one thing that attracts attention, to the advantage of the exhibitor 
of the plans and specifications used in the competition bids. The Patent 
Office rules also call for a border line. The size of the sheet of pure 
white paper on which a drawing is made must be exactly lox 15 inches. 
One inch from its edge a single marginal line is to be drawn leaving 
" the sight " precisely 8x 13 inches. 



424 



ROGERS' DRAWING AND DESIGN. 



as in drawingf a horizontal line, and straig^hten the 
paper so that its upper edge will be paralUl to the 
edge of the tee square blade. Pull the corner diag- 
onally opposite that in which the thumb-tack was 
placed, so as to stretch the paper slightly and push 
in another thumb-tack. Proceed in the same man- 
ner for the remaining two corners. 

The thumb-tacks or drawing-pins should have a 
head as thin as possible without cutting at its edgesj 
slightly concave on the underside next to the paper, 
and should be only so much convex on its upper 
side as will give it sufficient thickness to enable the 
pin to be secured to it; it is better to use four or 
more small pins along the edge of a sheet of paper 
than use one much larger pin at each corner. 

For particular work it is necessary to stretch the 
paper while it is damp. For stretching the paper 
in this way moisten the whole sheet on the under 
side, with the exception of a margin all around the 
sheet, of about half an inch and paste the dry border 
to the drawing board. To do this properly requires 
a certain amount of skill, and paper thus stretched 
gives undoubtedly a smoother surface than can be 
obtained when using thumb-tacks, but there are 
objections to this process as the paper stretched in 
this way is under a certain strain and may have some 
effect on the various dimensions of the drawing, 
when cut off the board. 



Once the drawing completed, cut the paper from 
the board with a knife, by following the lines previ- 
ously drawn all around the sheet for trimming. 
Make a continuous cut all around ; if one of the 
longer sides is cut first and then the opposite side 
there is danger of tearing the paper when cutting the 
remaining sides. 

PENCILING. 

The pencil drawing should look as nearly like the 
ink drawing as possible. A good draughtsman 
leaves his work in such a state that any competent 
person can without difficulty ink in what he has 
drawn. 

The pencil should always be drawn, not pushed. 
Lines are generally drawn from left to right and 
from the bottom to the top or upwards. Pencil lines 
should not be any longer than the proposed ink 
lines. By keeping a drawing in a neat, clean condi- 
tion when penciling, the use of the rubber upon the 
finished inked drawing will be greatly diminished. 

INKING. 

A drawing should be inked in only after the pen- 
ciling is entirely completed. Always begin at the top 
of the paper,first inking in all small circles and curves, 
then the larger circles and curves, next all horizontal 
lines, commencing again at the top of the drawing 



II 



ROGERS' DRAWING AND DESIGN. 



425 



and working downward. Then ink in all vertical 
lines, startinof on the left and moving- toward the 
right ; finally draw all oblique lines. 

Irregular curves, small circles and arcs are inked in 
first, because it is easier to draw a straight line up to 
a curve than it is to take a curve up to a straight line. 

DRAWING TO SCALE. 

The meaning of this is, that the drawing when 
done bears a definite proportion to the full size of 
the particular part, or, in other words, is precisely 
the same as it would appear if viewed through a 
diminishinof orlass. 

When it is required to make a dravving to a re- 
duced scale, that is, of a smaller size than the actual 
size of the object, say for instance, J^ full size, every 
dimension of the object in the drawing must be one- 
half the actual size ; in this case one inch on the 
object would be represented by 3^ inch. Such a 
reduced drawing could be made with an ordinary 
rule, this, however, would require every size of the 
object to be divided by the proportion of the scale, 
which would entail a very great loss of time in cal- 
culations. This can be avoided by simply dividing 
the rule itself by 2, from the beginning. Such a 
rule, or scale as it is generally called, will be divided 
in yi inches, each half inch representing one full inch 



divided into yi, j^, yi, ■^, each of these representing 
the same proportions of the actual sizes of the object 
to be drawn. From this contracted scale the dimen- 
sions and measurements are laid off on the drawing. 

A quarter size scale is made by taking three inches 
to represent one foot. Each of the three inches 
will be divided into 12 parts representing inches, 
each one of these again will be divided in i^, y%, 
■j?^, etc.; each one of these representing to a quarter 
size scale the actual sizes of ^, %,yi,~^ of an inch. 

It must be mentioned that in several instances, in 
this work, distances in one figure are said to be 
equal to corresponding distances in the same object 
in another view, while by actual measurement they 
are somewhat different; this is owing to the use of 
different scales — each scale separate should be 
marked on the drawing. 

Paper scales for large drawings are extremely use- 
ful and remarkably accurate. The advantage they 
possess over other kinds is that they expand and 
contract equally with the drawing paper during the 
various changes of the weather. 

The nickel-plated sheet-metal steel scale which has 
two graduated edges conduces to most accurate 
work; this instrument having only two scales the 
annoyance experienced of frequently turning it, is 
greatly reduced. 



426 



ROGERS' DRAWING AND DESIGN. 



A fiat boxwood scale with beveled edges has less 
pitch on its side and for that reason can be more 
quickly and easily read than others. 

SELECTION OF INSTRUMENTS. 

The choice of drawing tools is one of the most 
difficult points to settle that can present itself. Suc- 
cess or failure may hang upon the getting the most 
suitable tools, hence it is well to follow the advice 
of some professional draughtsman, and in buying, 
procure such tools as are immediately needed and to 
add others as occasion demands. 

The best quality of instruments last longer and in 
the end are the cheapest. German silver is the best 
metal used, much better than brass ; the use of 
pocket or folding instruments is to be avoided ; if it 
is necessary to carry the instruments nothing is 
better than to fold them up in a piece of chamois 
leather, or to have a little satchel or grip which will 
also accommodate the triangles, ink, colors, etc. 

Louis Rouillon, B. S., Instructor of Drawing in 
Pratt Institute, New York, recommends the follow- 
ing set of tools for the beginner : 

Compasses, 5^ inches, with needle point; pen, 
pencil and lengthening bar. 

Drawing pen, 4^ inches. 



T square, 24-inch blade. 

45-degree triangle, 9 inches. 

30 and 60-degree triangle, g inches. 

I Scroll. 

Dixon's V. H. pencil. 

12-inch boxwood scale, flat, graduated 1-16 inch 
the entire length. 

Bottle of liquid India ink, four thumb-tacks, pen 
and ink eraser. 

20 sheets drawing paper, 11 X 15 inches, and a 
drawing-board about 16 x 23 inches will also be 
necessary. 

Henry Raabe, M. E., is entitled to credit for the 
following list of instruments : 

I Pair of compasses, with pencil, pen, needle point, 
and lengthening bar ; i Pair of dividers ; i Draw- 
ing pen ; 1 Bow pen ; i Bow pencil ; i Bow divid- 
ers ; I 45-degree triangle ; i 60-degree by 30-degree 
triangle ; i Tee square ; i Drawing board ; i Pro- 
tractor; I Scale from 1" to the foot to /^ " to the 
foot ; I Scale from 3" to the foot to ^" to the foot; 
I Pencil rubber; i Ink eraser; i Pen holder with 
pens ; i Pencil holder for short pencils ; Compass 
pencils ; Pencils from 6 H. to 3 H. (drawing pen- 
cils) ; Pencil pointer ; Drawing ink ; Sketch pads ; 
Sketch pencils (soft); Thumb tacks, paper and trac- 



ing cloth. 



PRACTICAL RULES AND USEFUL DATA, 



For mental drill there is nothing better than the solution of mathematical problems. It is not 
necessary that these problems be intricate and in the higher branches, but only not so easy as to be 
readily understood without active and sustained brain work. 

Accuracy, first of all, rapidity and a familiarity with the elements of numbers and their application 
to the problems immediately surrounding one, — these are the foundations of many successful lives ; to 
most minds the study of mathematics is dry and uninteresting ; to make the subject acceptable it must 
be presented in such a form as to immediately appeal to the student as of great practical value. This 
value is proven when applications are made to problems that confront the draughtsman and engineer in 
his daily routine. There is no more interesting subject for one who is disposed to study than that of 
useful numbers. It literally opens a new world to the student. It gives him his first idea of what it 
means to really /rciz;^ anything, for the demonstrations of figures and geometry prove absolutely and 
completely the propositions with which they deal. 

"In the wide expanse of mathematics it has been a task of the utmost difficulty for the author to 
lay out a road that would not too soon weary or discourage the student ; if he had his wish he would 
gladly advance step by step with his pupil, and much better explain, byword and gesture and emphasis, 
the great principles which underlie the operations of mechanics ; to do this would be impossible, so he 
writes his admonition ia two short words: In case of obstacles, 'go on.' If some rule or process seetUS 
too hard to learn, go around the difficulty, always advancing, and, in time, retu7'7z and conquer." 

The foregoing paragraphs are simply to emphasize a few words explaining the value of the tallies 
which are printed in the following pages ; tables of the results of mathematical calculations are of 
immense economy in time, in guaranteeing accuracy and the saving of much drudgery 

To thoroughly understand the easy and helpful uspof the tables which follow should be the pleasant 
task of the student ; the value of a teacher or instructor at this point cannot be over estimated ; men are 
not made to do their work alone, to help and to be helped is the universal law ; when assistance is to 
be had whether it is for pay or favor the student should avail himself of it with many thanks. 

429 



430 



ROGERS' DRAWING AND DESIGN. 



ELEMENTS OF ALGEBRA. 

Algebra Is a mathematical science which teaches 
the art of making calculations by letters and signs 
instead of figures. 

The name comes from two Arabic words, algabron, 
reduction of parts to a whole. 

The letters and signs are called Symbols. 

Quantities in algebra are expressed by letters, or 
by a combination of letters and figures ; as a, b, c, 
2.^, Zy, S'S, etc. 

The first letters of the alphabet are used to ex- 
press known quantities ; the last letters, those which 
are unknown. 

The Letters employed have no fixed numerical 
value of themselves. Any letter may represent any 
number, and the same letter may represent different 
numbers, but in each sum the same letter must 
always stand for the same amount. 

The operations to be performed are expressed by 
the same signs as in Arithmetic; thus + means 
Addition, — expresses Subtraction, and X stands 
for Multiplication. 

Thus « + /; denotes the sum of a and b and is read 



a plus b ; a — b means a less b; and aV^b shows that 
a and b are to be multiplied together. 

Multiplication is also denoted by a period between 
the factors as a.(5. But the multiplication of letters 
is more commonly expressed by writing them to- 
gether, the signs being omitted. 

Example : 7 abc is the same as 'j'XaXbXc. 

The sign of Division is -^, thus a^k' is read a 
divided by b ; but this is also expressed -r-/ the sign 
of Equality is two short horizontal lines as a^^=b and 
is read a equals b. 

The Parenthesis ( ) or Vinculum , indicates 

that the included quantities are taken collectively or 
as one quantity. 

Example : 3 {a-\-S) and 3«+<5 each denote that 
the sum of a and b is multiplied by 3. 

The character . " . denotes hence, therefore. 

A Coefficient is a number or letter prefixed to a 
quantity, to show how many times the quantity is to 
be taken. Hence a coefficient is a multiplier or 
factor ; thus in 5«, 5 is a numeral coefficient of a. 

When no numeral coefficient is expressed, i is 
always understood. Thus xy means \xy. 



ROGERS' DRAWING AND DESIGN. 



431 



DEFINITIONS. 

An Algebraic Operation is combining quantities 
according to the principles of algebra. 

A Theorem is a statement of a principle to be 
proved. 

A Problem is something proposed to be done, as 
a question to be solved. 

The Expression of Equality between two quanti- 
ties is called an Equation. 

An Algebraic Expression is any quantity expressed 
in algebraic language, as yi, ^a — ']a, etc. 

The Terms of an algebraic expression are those 
parts which are connected by the signs + and — . 

Thus in aArb there are two terms ; \nx, y and z — a 
there are three. 

A Positive Quantity is one that is to be added and 
has the sign + prefixed to it, as 4^ + 33. 

A Negative Quantity is one that is to be subtracted 
and has the sign — prefixed to it, as \a — 3^. 

A Simple Quantity is a single letter, or several 
letters written together without the sign + or — , as 
a, ab, 3Xji'. 

A Compound Quantity is two or more simple 
quantities connected by the sign + or — , as 2,a-\- i,b, 
ix — y. 

The Axioms in algebra are self-evident truths as 
exemplified on pages 85 and 86. 



ADVANTAGES OF ALGEBRA. 

In algebra numbers are expressed by the letters 
of the alphabet ; the advantage of the substitu- 
tion is that we are enabled to pursue our investiga- 
tions without being embarrassed by the necessity of 
performing arithmetical operations at every step. 

Thus, if a given number be represented by the 
letter a, we know that 2a will represent twice that 
number, and ^^ the half of that number, whatever 
the value of a may be. In like manner if a be taken 
from a there will be nothing left and this result will 
equally hold whether a be 5, or 7, or 1000, or any 
other number whatever. 

By the aid of algebra, therefore, we are enabled to 
analyze and determine the abstract properties of 
numbers, and we are also enabled to resolve many 
questions that by simple arithmetic would either be 
difficult or impossible. 

A draughtsman or engineer has but little practical 
use for a too extended acquaintance with algebra, as 
nearly all the algebraic rules have been transferred 
to ordinary arithmetical computation, but as the 
algebraic system is so inwoven into the school and 
college course of instruction it is well for every one 
to know something- of the elements of the science. 

Arithmeticians for very many years have made a 
study of the use of formulce (this is Latin for the 



432 



ROGERS' DRAWING AND DESIGN. 



word forms) in stating problems and rules ; these 
forms are nearly all expressed in algebraic terms, 
The advantage to be derived from the use of these 
is that it puts into a short space what otherwise 
might necessitate the use of a long verbal or written 
explanation. 

Another advantage is that the memory retains the 
form of the expression much easier ?.nd longer than 
the longer method of expression, and it may be re- 
marked that those who once become accustomed to 
the use of formulae seldom abandon their employment. 
Examples Explaining the Solvng of Formul.k 

1. \{ X ^ a + b — ^ + d — f ; what must be the 
value of X when «= lo, <5=7, ^==9, «?^4, and 
/=6? 

First substitute the figures for the letters, thus : — 
A' = 10 + 7 — 9+4 — 6, then proceed as in the 
Arithmetical part. 

X ^ 21 — 15 = 6 Answer. 

2. If .r == 4 g-\- 2 m — 7 11 — p + 3 ^ / find the 

= 6 ; /> == I ; and 



J ' 



value of X when ^= 5 ; m- 

Here 4_^= 4 times 5 ^ 20 ; 2 m -^ twice 3^6; 
■J n= y times 6 == 42 ; and 2) ^ '^ i times 8 = 24; 
Hence, ^ = 20 + 6 — 42 — i +24 

= 50—43 
= 7 Answer. 



U X- 



%d+\c- 



Va./ ; find the value 



of .;ir when a = 10 ; d=2a^; ^^25; and/"^ 12. 

As «= 10, then ^ «=^5 ; as d=^ 24, then y^d=^ 
6 ; as ^ = 25, then \ c =5 ; and as f^^ 1 2, then ^ 

Hence, .1-^ 5 — 6 + 5 — 9. 
= 10 — 15. 
= — 5 Answer. 

4. If X = c — (I — f) ; find the value of jirwhen ^^^8. 
-y = 3/^ and / ^ I ^ 

x^^ — (^ — i^); here 3!/^ is divided 
by 2= i^. 
= 8-(i3^-i>^) 
= 8-i< 
= 7^ Answer. 

^. X ^ a b + c d — e f ; where « ^ 2, <5 = 3, c 
= 4, d = s, e = 6, and/= 7. 

;t: = 2X3 + 4X5 — 6X7 
= 6 + 20 — 42 
= 26 — 42 
= — 16 Answer. 

AB 
6. Then {{ x = j-. ^ ; what is the value of x 

when A^6; B = 7; C=io; and D = 16 ? 

6X7 42 

x^ -? = ~7- = 7 Answer. 

16 — 10 6 ' 



ROGERS' DRAWING AND DESIGN. 



438 



LOGARITHMS. 

This word is composed of two Greek nouns' 
meaning reason and mimber ; a logarithm is an 
artificial number so related to the natural numbers 
that the multiplication and division of the latter 
may be performed by addition and subtraction and 
by their use the much more difificult operations of 
raising to powers and the extraction of roots are 
effected by easy cases of multiplication and division. 

The early computers of logarithms carried them to 
ten places of decimals, but it was soon found that five 
and seven places were sufficient for most purposes ; 
those given in this book are carried to six places. 
Naperian logarithms are called natural and also 
Hyperbolic logarithms ; common logarithms are 
called the decimal, and also the Briggsian System. 

In the Table, letter N over the first column stands 
for "number" ; after loo (see page 436) the num- 
bers at the top of the columns express the tenth 
parts of N. 

Note. — Logarithms were invented and a table published in 1614 
by John Napier, of Scotland ; but the kind now chiefly in use were pro- 
posed by his contemporary Henry Briggs, of London. The first extended 
table of common logarithms were calculated by Adrian Vlacq in 1628, 
and have been the basis of every one since published ; when logarithms 
are spoken of without any qualifications common logarithms are to be 
understood. The labor of the operation incurred in the ordinary pro- 
cesses of arithmetic is often enormous ; by the use of logarithms this 
labor is greatly lessened ; logarithms are of inestimable value in the so- 
called higher mathematics, in navigation, in surveying, and in the inves- 
tigation of many problems in physics. 



LOGARITHMIC TABLE. 

When the engineer or draughtsman is required 
to make long and difficult calculations, consisting of 
the multiplication, division, squaring, etc., of num- 
bers, the logarithmic table will, as explained in the 
note, be of such assistance as may amply repay the 
study of the subject and the acquirement of rapid 
and accurate use of the table. 

It must be understood that but an outline only of 
this interesting study is here presented and that the 
columns of figures given in the tables beginning on 
page 435 are but a very small part of those published 
in advanced works on mathematics ; hence the ex- 
amples given of the use of the table are necessarily 
confined to very small numbers. 

To use the table, fi.nd the number in the first 
column marked N, and in the next column the corre- 
sponding logarithm, will be found. 

The figures given in the column are only the 
decimal part of the logarithm. The rules and ex- 
amples for the application of logarithms are as fol- 
lows : 

Rule : To m.ultiply tzvo numbers, add their log- 
arithms, a7id the result will be the logarithm of the 
product. 



434 



ROGERS' DRAWING AND DESIGN. 



Example: Mul 


tiply 25 by 6 


log. 25 
log. 6 


= I -397940 

■778151 




2. 1 7609 1 


Proof: 25 
6 





log. 150 



• 50 

Rule : To divide one number by another one, sub- 
tract tlie logarithm of the divisor from the logarithm 
of the dividend. 

Example : Divide 1 75 by 7 

log. 175 =" 2,243038 
log. 7 = -845098 

1.397940-= log. 25 

Proof : 175 -^ 7 ^= 25 

Rule . To find aiiy power of any number, multi- 
ply the index of the power with the logarithm of the 
number ; the product is the logarithm of the power. 

Example : Find the value of 3* 
log. 3 = .477121 
4 



Proof : 3 X 



1.908484 = log. 81 



9X3 = 27 X 3 



F-XAMPLE : Find the value of 14^ 
log. 14 = 1. 146 1 28 



2.292256 = log. 196 

Rule : To find any root of a number, divide the 
logarithm of the num,ber by the index of the root. 

Example : Find the value of V64 

log. 64= 1.806 1 80 -4- 2 =.903090)'== log. 8. 



Example: Find the value of V 128 

log. 128 = 2.107210 -H 7 = .301030 = log. 2. 

To find the charactistic or whole number to be 
placed before the mantissa, or decimal part of the 
logarithm, proceed as follows : 

Rule : // the nmnber is between i and 10 the 
logarithm is only a fraction. The logarithm 0/ 
JO is I, between 10 and 100, a i has to be placed in 
front of the fractional part found in the table ; be- 
tween J 00 and 1,000, a 2 forms the whole number ; 
between 1,000 and 10,000 the figure is j, and so on. 

Example: What is the logarithm of 123? — by 
looking in the table we find log. 123 = ,089905 and 
placing a 2 in front of the decimal point, we have 
for the true log. of 123 ^ 2.089905. 



ROGERS' DRAWING AND DESIGN. 



435 



TABLE OF LOGA- 
RITHMS. 

There are two different 
tables of logarithms in use, 
one is called the Napierian 
system, named after its in- 
ventor, and the common 
system of which the base' is 
lo; the accompanying ta- 
bles are common loga- 
rithms. 

The logarithm of a num- 
ber usually consists of two 
parts, the integral, or whole 
part, and a fractional part ; 
the integral part is called 
the characteristic or index, 
and the fractional part the 
mantissa. The last word is 
from the Latin and means 
an addition. 

The abbrviviation of the 
words "logarithm of" is log. 
or Log., thus: log. 136 = 
2-133539, the characteristic 
of the logarithm i ^6 beine 
2, and the mantissa .133539. 
See Table, page 437. 

I n the tables the mantissas 
only are given. 



N 


Log. 


N 


Log. 


N 


Log. 


N 


Log. 


1 


000000 


26 


414973 


51 


707570 


76 


880814 


2 


301030 


27 


431364 


52 


716003 


77 


886491 


3 


477121 


28 


447158 


53 


724276 


78 


892095 


4 


602060 


29 


462398 


54 


732394 


79 


897627 


5 


698970 


^30 


477121 


55 


740363 


80 


903090 


6 


778151 


31 


491362 


56 


748188 


81 


908485 


7 


845098 


32 


505150 


57 


755875 


82 


913814 


8 


903090 


33 


518514 


58 


763428 


83 


919078 


9 


954243 


34 


531479 


59 


770852 


84 


924279 


10 


000000 


35 


544068 


60 


778151 


85 


929419 


11 


041393 


36 


556303 


61 


785330 


86 


934498 


12 


079181 


37 


568202 


62 


792392 


87 


939519 


13 


113943 


38 


579784 


63 


799341 


88 


944483 


14 


146128 


39 


591065 


64 


806180 


89 


949390 


15 


176091 


40 


602060 


65 


812913 


90 


954243 


16 


204120 


41 


612784 


66 


ai9544 


91 


959041 


17 


230449 


42 


623249 


67 


826075 


92 


963788 


18 


255273 


43 


633468 


68 


832509 


93 


968483 


19 


278754 


44 


643453 


69 


838849 


94 


973128 


20 


301030 


45 


653213 


70 


845098 


95 


977724 


21 


322219 


46 


662758 


71 


851258 


96 


982271 


22 


342423 


47 


672098 


72 


857332 


97 


986772 


23 


361728 


48 


681241 


73 


863323 


98 


991226 


24 


380211 


49 


690196 


74 


869232 


99 


995635 


25 


397940 


50 


698970 


75 


875061 


100 


000000 



436 






ROGERS- DRAWING 


AND DESIGN. 






. 


=1 










TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




100 


000000 


0U0434 


000868 


001301 


001734 


002166 


002598 


003029 


003461' 


003891 




101 


004321 


004751 


005181 


005609 


006038 


006466 


006894 


007321 


007748 


008174 






102 


008600 


009026 


009451 


009876 


010300 


010724 


011147 


011570 


011993 


012415 






103 


012837 


013259 


013680 


014100 


014521 


014940 


015360 


015779 


016197 


016616 


1 




104 


017033 


017451 


017868 


018284 


018700 


019116 


019532 


019947 


020361 


020775 






105 


02.1189 


021603 


022016 


022428 


022841 


023252 


023664 


024075 


024486 ! 


024896 






106 


025306 


025715 


026125 


026533 


026942 


027350 


027757 


028164 


028571 ^ 


\ 028978 






107 


029384 


029789 


030195 


030600 


031004 


031408 


031812 


032216 


032619 


033021 


1 




108 


033424 


033826 


034227 


034628 


035029 


035430 


035830 


036230 


036629 


037028 






109 


037426 


037825 


038223 


038620 


039017 


039414 


039811 


040207 


040602 


040998 






110 


041393 


041787 


042182 


042576 


042969 


043362 


043755 


044148 


044540 


044932 






111 


045323 


045714 


046105 


046495 


046885 


047275 


047664 


048053 


048442 


048830 






112 


049218 


049606 


049993 


050380 


050766 


051153 


051538 


051924 


052309 


052694 






113 


053078 


053463 


053846 


054230 


054613 


054996 


055378 


055760 


056142 


056524 






114 


056905 


057286 


057666 


058046 


058426 


058805 


059185 


059568 


059942 


060320 






115 


060698 


061075 


061452 


061829 


062206 


062582 


062958 


063333 


063709 


064088 






116 


064458 


064832 


065206 


065580 


065953 


066326 


066699 


067071 


067443 


067815 






117 


068186 


068557 


068928 


069298 


069668 


070038 


070407 


070776 


071145 


071514 






118 


071882 


072250 


072617 


072985 


073352 


073718 


074085 


074451 


074816 


075182 






119 


075547 


075912 


076276 


076640 


077004 


077368 


077731 


078094 


078457 


078819 






120 


079181 


079543 


079904 


080266 


080626 


080987 


081347 


081707 


082067 


082426 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 





























ROGERS' DRAWING AND DESIGN. 



437 









TABLE OF 


LOGARITHMS— Continued. 








N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


121 


082785 


083144 


083503 


083861 


084219 


084576 


084934 


085291 


085647 


086004 


122 


086360 


086716 


087071 


087426 


087781 


088136 


088490 


088845 


089198 


089552 


123 


089905 


090258 


090611 


090963 


091315 


091.667 


092018 


092370 


092721 


093071 


121 


093422 


093772 


094122 


094471 


094820 


095169 


095518 


095866 


096215 


096562 


125 


096910 


097257 


097604 


097951 


098298 


098644 


098990 


099335 


099681 


100026 


126 


100371 


100715 


101059 


101403 


101747 


102091 


102434 


102777 


103119 


103462 


127 


103804 


104146 


104487 


104828 


105169 


105510 


105851 


106191 


106531 


106871 


128 


107210 


107549 


107888 


108227 


108565 


108903 


109241 


109579 


109916 


110253 


129 


110590 


110926 


111263 


111599 


111934 


112270 


112605 


112940 


113275 


113609 


130 


113943 


114277 


114611 


114944 


115278 


115611 


115943 


116276 


116608 


116940 


131 


117271 


117603 


117934 


118265 


118595 


118926 


119256 


119586 


119915 


120245 


132 


120574 


120903 


121231 


121560 


121888 


122216 


122544 


122871 


123198 


123525 


133 


123852 


124178 


124504 


124830 


125156 


125481 


125806 


126131 


126456 


126781 


134 


127105 


127429 


127753 


128076 


128399 


128722 


129045 


129368 


129690 


130012 


136 


130334 


130655 


130977 


131298 


J31619 


131939 


132260 


132580 


132900 


133219 


136 


i 133539 


133858 


134177 


134496 


134814 


135133 


135451 


135769 


136086 


136403 


137 


1 136721 


137037 


137354 


137671 


137987 


138303 


138618 


138934 


139249 


139564 


138 


1 139879 


140194 


140508 


140822 


141136 


141450 


141763 


142076 


142389 


142702 


139 


143015 


143327 


143639 


143951 


144263 


144574 


144885 


145196 


145507 


145818 


140 


146128 


146438 


146748 


147058 


147367 


147676 


147985 


148294 


148603 


148911 


N 


1 


1 


2 


3 


4 


6 


6 


7 


8 


9 



438 



ROGERS' DRAWING AND DESIGN. 









TABLE OF 


LOGARITHMS-Continued. 








N 





1 


2 


3 


4 


5 


6 


f 


S 


9 


141 


149219 


149527 


149835 


150142 


150449 


150756 


151063 


151370 


151676 


151982 


142 


152288 


152594 


152900 


153205 


153510 


153815 


154120 


154424 


154728 


155032 


143 


155336 


155640 


155943 


156246 


156549 


156852 


157154 


157457 


157759 


158061 


144 


158362 


158664 


158965 


159266 


159567 


159868 


160168 


160469 


160769 


161068 


145 


161368 


161667 


161967 


162266 


162564 


162863 


163161 


163460 


163758 


164055 


146 


164353 


164650 


164947 


165244 


165541 


165838 


166134 


166430 


166726 


' 167022 


147 


167317 


167613 


167908 


168203 


168497 


168792 


169086 


169380 


169674 


169968 


148 


170262 


170555 


170848 


171141 


171434 


171726 


172019 


172311 


172603 


172895 


149 


173186 


173478 


173769 


174060 


174351 


174641 


174932 


175222 


175512 


175802 


160 


176091 


176381 


176670 


176959 


177248 


177536 


177825 


178113 


178401 


178689 


151 


178977 


179264 


179552 


179839 


180126 


180413 


180699 


180986 


181272 


181558 


152 


181844 


182129 


182415 


182700 


182985 


183270 


183555 


183839 


184123 


184407 


153 


184691 


184975 


185259 


185542 


185825 


186108 


186391 


186674 


186956 


187239 


154 


187521 


187803 


188084 


188366 


188647 


188928 


189209 


189490 


189771 


190051 


165 


190332 


190612 


190892 


191171 


191451 


191730 


192010 


192289 


192567 


192846 


156 


193125 


193403 


193681 


193959 


194237 


194514 


194792 


195069 


195346 


195623 


157 


195900 


196176 


196453 


196729 


197005 


197281 


197556 


197832 


198107 


198382 


158 


198657 


198932 


199206 


199481 


199755 


200029 


200303 


200577 


200850 


201124 


169 


201397 


201670 


201943 


202216 


202488 


202761 


203033 


203305 


203577 


203848 


160 


2041^0 


204391 


204663 


204934 


205204 


205475 


205746 


206016 


206286 


206556 


N 





1 


2 


3 


4 


6 


6 


7 


8 


9 









ROGERS' DRAWING 


AND DESIGN. 






439 










TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




161 


206826 


207096 


207365 


207634 


207904 


208173 


208441 


208710 


208979 


209247 




162 


209515 


209783 


210051 


210319 


210586 


210853 


211121 


211388 


211654 


211921 






163 


212188 


212454 


212720 


212986 


213252 


213518 


213783 


214049 


214314 


214579 






164 


214844 


215109 


215373 


215638 


215902 


216166 


216430 


216694 


216957 


217221 


i 




165 


217484 


217747 


218010 


218273 


218536 


218798 


219060 


219323 


219585 


219846 






166 


220108 


220370 


220631 


220892 


221153 


221414 


221675 


221936 


222196 


222456 






167 


222716 


222976 


223236 


223496 


223755 


224015 


224274 


224533 


224792 


225051 






168 


225309 


225568 


225826 


226084 


226342 


226600 


226858 


227115 


227372 


227630 




1 


169 


227887 


228144 


228400 


228657 


228913 


229170 


229426 


229682 


229938 


230193 


1 




170 


230449 


230704 


230960 


231215 


231470 


231724 


231979 


232234 


232488 


232742 






171 


232996 


233250 


233504 


283757 


234011 


234264 


234517 


234770 


235023 


235276 






172 


235528 


235781 


236033 


236285 


236537 


236789 


237041 


237292 


237544 


237795 






173 


238046 


238297 


238548 


238799 


239049 


239299 


239550 


239800 


240050 


240300 


! 




174 


240549 


240799 


241048 


241297 


241546 


241795 


242044 


242293 


242541 


242790 






175 


243038 


243286 


243534 


243782 


244030 


244277 


244525 


244772 


245019 


245266 


1 




176 


245513 


245759 


246006 


246252 


246499 


246745 


246991 


247237 


247482 


247728 






177 


247973 


248219 


248464 


248709 


248954 


249198 


249443 


249687 


249932 


250176 






176 


250420 


250664 


250908 


251151 


251395 


251638 


251881 


252125 


252368 


252610 




1 


179 


252853 


253096 


253338 


253580 


253822 


254064 


254306 


254548 


254790 


255031 


i 




180 


255273 


255514 


255755 


255996 


256237 


256477 


256718 


256958 


257198 


257439 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 





























440 






ROGERS' DRAWING 


AND DESIGN. 












- 




TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




181 


257679 


257918 


258158 


258398 


258637 


258877 


259116 


259355 


259594 


259833 




182 


260071 


260310 


260548 


260787 


261025 


261263 


261501 


261739 


261976 


262214 






183 


262451 


262688 


262925 


263162 


263399 


263636 


263873 


264109 


264346 


264582 






184 


264818 


265054 


265290 


265525 


265761 


265996 


266232 


266467 


266702 


266937 






185 


267172 


267406 


267641 


267875 


268110 


268344 


268578 


268812 


269046 


269279 






186 


269513 


269746 


269980 


270213 


270446 


270679 


270912 


271144 


271377 


271609 






187 


271842 


272074 


272306 


272538 


272770 


273001 


273233 


273464 


273696 


273927 






188 


274158 


274389 


274620 


274850 


275081 


275311 


275542 


275772 


276002 


276232 






189 


276462 


276692 


276921 


277151 


277380 


277609 


277838 


278067 


278296 


278525 






190 


278754 


278982 


279211 


279439 


279667 


279895 


280123 


280351 


280578 


280806 






191 


281033 


281261 


281488 


281715 


281942 


282169 


282396 


282622 


282849 


283075 






192 


283301 


283527 


283753 


283979 


284205 


284431 


284656 


284882 


285107 


285332 






193 


285557 


285782 


286007 


286232 


286456 


286681 


286905 


287130 


287354 


287578 






194 


287802 


288026 


288249 


288473 


288696 


288920 


289143 


289366 


289589 


289812 






195 


290035 


290257 


290480 


290702 


290925 


291147 


291369 


291591 


291813 


292034 






196 


292256 


292478 


292699 


292920 


293141 


293363 


293584 


293804 


294025 


294246 






197 


294466 


294687 


294907 


295127 


295347 


295567 


295787 


296007 


296226 


296446 






198 


296665 


296884 


297104 


297323 


297542 


297761 


297979 


298198 


298416 


298635 


1 




199 


298853 


299071 


299289 


299507 


299725 


299943 


300161 


300378 


300595 


300813 


1 


200 


301030 


301247 


301464 


301681 


301898 


302114 


302331 


302547 


302764 


302980 


1 

— 1 


N 





1 


2 


3 


4 


5 


6 


7 


8 


» 

































ROGERS' DRAWING 


AND DESIGN. 






441 










TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


1 

8 


9 




201 


303196 


303412 


303628 


303844 


304059 


304275 


304491 


304706 


304921 


305136 




202 


305351 


305566 


305781 


305996 


306211 


306425 


306639 


306854 


307068 


307282 






203 


307496 


307710 


307924 


308137 


308351 


308564 


308778 


308991 


309204 


309417 






204 


309630 


309S43 


310056 


310268 


310481 


310693 


310906 


311118 


311330 


311542 






205 


311754 


311966 


312177 


312389 


312600 


312812 


313023 


313234 


313445 


313656 






206 


313867 


314078 


314289 


314499 


314710 


314920 


315130 


315340 


315551 


315760 






207 


315970 


316180 


316390 


316599 


316809 


317018 


317227 


317436 


317646 


317854 




; 


208 


318063 


318272 


318481 


318689 


318898 


319106 


319314 


319522 


319730 


319938 






209 


320146 


320354 


320562 


320769 


320977 


321184 


321391 


321598 


321805 


322012 






210 


322219 


322426 


322633 


322839 


323046 


323252 


323458 


323665 


323871 


324077 






211 


324282 


324488 


324694 


324899 


325105 


325310 


325516 


325721 


325926 


326131 






212 


326336 


326541 


326745 


326950 


327155 


327359 


327563 


327767 


327972 


328176 






213 


328380 


328583 


328787 


328991 


329194 


329398 


329601 


329805 


330008 


330211 






214 


330414 


330617 


330819 


331022 


331225 


331427 


331630 


331832 


332034 


332236 






215 


332438 


332640 


332842 


333044 


333246 


333447 


333649 


333850 


334051 


334253 






216 


334454 


334655 


334856 


335057 


335257 


335458 


335658 


335859 


336059 


336260 






217 


336460 


336660 


336860 


337060 


337260 


337459 


337659 


337858 


338058 


338257 






218 


338456 


338656 


338855 


339034 


339253 


339451 


339650 


339849 


340047 


340246 






219 


340444 


340642 


340841 


341039 


341237 


341435 


341632 


341830 


342028 


342225 






220 


342423 


342620 


342817 


343014 


343212 


343409 


343606 


343802 


343999 


344196 
9 




N 





1 


2 


3 


4 


5 


6 


7 


8 





























442 






ROGERS' DRAWING 


AND DESIGN. 






- 










TABLE OF LOGARITHMS-Continoed. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




221 


344392 


344589 


344785 


344981 


345178 


345374 


345570 


345766 


345962 


346157 




222 


346353 


346549 


346744 


346939 


347135 


847330 


347525 


347720 


347915 


348110 


^ 




223 


348305 


348500 


348694 


348889 


349083 


349278 


349472 


349666 


349860 


350054 






224 


350248 


350442 


350636 


350829 


351023 


351216 


351410 


351603 


351796 


351989 






225 


352183 


352375 


352568 


352761 


352954 


353147 


353339 


353532 


353724 


353916 






226 


354108 


354301 


354493 


354685 


354876 


355068 


355260 


355452 


355643 


355834 






227 


356026 


356217 


356408 


356599 


356790 


356981 


357172 


357363 


357554 


357744 






228 


357935 


358125 


358316 


358506 


358696 


358886 


359076 


359266 


359456 


359646 






229 


359835 


360025 


360215 


360404 


360593 


360783 


360972 


361161 


361350 


361539 






230 


361728 


361917 


362105 


362294 


362482 


362671 


362859 


363048 


363236 


363424 






231 


363612 


363800 


363988 


364176 


364363 


364551 


364739 


364926 


365113 


365301 






232 


365488 


, 365675 


365862 


366049 


366236 


366423 


366610 


366796 


366983 


367169 






233 


367356 


367542 


367729 


367915 


368101 


368287 


368473 


368659 


368845 


369030 






234 


369216 


369401 


369587 


369772 


369958 


370143 


370328 


370513 


370698 


370883 






235 


371068 


371253 


371437 


371622 


371806 


371991 


372175 


372360 


372544 


372728 






236 


372912 


373096 


373280 


373464 


373647 


373831 


374015 


374198 


374382 


374565 






237 


374748 


374932 


375115 


375298 


375481 


375664 


375846 


376029 


376212 


376394 






238 


376577 


376759 


376942 


377124 


377306 


377488 


377670 


377852 


378034 


378216 






239 


378398 


378580 


378761 


378943 


379124 


379306 


379487 


379668 


379849 


380030 






240 


380211 


380392 


380573 


380754 


380934 


381115 


381296 


381476 


381656 


381837 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 

































ROGERS' DRAWING 


AND DESIGN. 






443 










TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




241 


382017 


382197 


382377 


382557 


382737 


382917 


383097 


383277 


383456 


383636 




242 


383815 


383995 


384174 


384353 


384533 


384712 


384891 


385070 


385249 


385428 






243 


385606 


385785 


385964 


386142 


386321 


386499 


386677 


386856 


387034 


387212 






244 


387390 


387568 


387746 


387923 


388101 


388279 


388456 


388634 


388811 


388989 






245 


389166 


389343 


389520 


389698 


389875 


390051 


390228 


390405 


390582 


390759 






246 


390935 


391112 


391288 


391464 


391641 


391817 


391993 


392169 


392345 


392521 




] 


247 


392697 


392873 


393048 


393224 


393400 


393575 


393751 


393926 


394101 


394277 






248 


394452 


394627 


394802 


394977 


395152 


395326 


395501 


395676 


395850 


396025 






249 


396199 


396374 


396548 


396722 


396896 


397071 


397245 


397419 


397592 


397766 






260 


397940 


398114 


398287 


398461 


398634 


398808 


398981 


399154 


399328 


399501 






261 


399674 


399847 


400020 


400192 


400365 


400538 


400711 


400883 


401056 


401228 






262 


401401 


401573 


401745 


401917 


402089 


402261 


402433 


402605 


402777 


402949 


1 




253 


403121 


403292 


403464 


403635 


403807 


403978 


404149 


404320 


404492 


404663 






264 


404834 


405005 


405176 


405346 


405517 


405688 


405858 


406029 


406199 


406370 






255 


406540 


406710 


406881 


407051 


407221 


407391 


407561 


407731 


407901 


408070 






256 


408240 


408410 


408579 


408749 


408918 


409087 


409257 


409426 


409595 


409764 




) 


257 


409933 


410102 


410271 


410440 


410609 


410777 


410946 


411114 


411283 


411451 






258 


411620 


411788 


411956 


412124 


412293 


412461 


412629 


412796 


412964 


413132 






259 


413300 


413467 


413635 


413803 


413970 


414137 


414305 


414472 


414639 


414806 






260 


414973 


415140 


415307 


415474 


415641 


415808 


415974 


416141 


416308 


416474 




N 





1 


2 


3 


4 


6 


6 


7 


8 


9 




I 

























444 






ROGERS' DRAWING 


AND DESIGN. 






- 










TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




261 


416641 


416807 


416973 


417139 


417306 


417472 


417638 


417804 


417970 


418135 




262 


418301 


418467 


418633 


418798 


418964 


419129 


419295 


419460 


419625 


419791 






263 


419956 


420121 


420286 


420451 


420616 


420781 


420945 


421110 


421275 


421439 






264 


421604 


421768 


421933 


422097 


422261 


422426 


422590 


422754 


422918 


423082 






265 


423246 


423410 


423574 


423737 


423901 


424065 


424228 


424392 


424555 


424718 




i 


266 


424882 


425045 


425208 


425371 


425534 


425697 


425860 


426023 


426186 


426349 




i 


267 


426511 


426674 


426836 


426999 


427161 


427324 


427486 


427648 


427811 


427973 






268 


428135 


428297 


428459 


428621 


428783 


428944 


429106 


429268 


429429 


429591 






269 


429752 


429914 


430075 


430236 


430398 


430559 


430720 


430881 


431042 


431203 






270 


431364 


431525 


431685 


431846 


432007 


432167 


432328 


432488 


432649 


432809 






271 


432969 


433130 


433290 


433450 


433610 


433770 


433930 


434090 


434249 


434409 






272 


434569 


434729 


434888 


435048 


435207 


435367 


435526 


435685 


435844 


436004 






273 


436163 


436322 


436481 


436640 


436799 


436957 


437116 


437275 


437433 


437592 






274 


437751 


437909 


438067 


438226 


438384 


438542 


438701 


438859 


439017 


439175 






275 


439333 


439491 


439648 


439806 


439964 


440122 


440279 


440437 


440594 


440752 






276 


440909 


441066 


441224 


441381 


441538 


441695 


441852 


442009 


442166 


442323 






277 


442480 


442637 


442793 


442950 


443106 


443263 


443419 


443576 


443732 


443889 






278 


444045 


444201 


444357 


444513 


444669 


444825 


444981 


445137 


445293 


445449 






279 


445604 


445760 


445915 


446071 


446226 


446382 


446537 


446692 


446848 


447003 






280 


447158 


447313 


447468 


447623 


447778 


447933 


448088 


448242 


448397 


448552 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



































ROGERS' DRAWING 


AND DESIGN. 






445 










TABLE OF ] 


LOGARITHMS-Contlnued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




281 


448706 


448861 


449015 


449170 


449324 


449478 


449633 


449787 


449941 


450095 




282 


450249 


450403 


450557 


450711 


450865 


451018 


451172 


451326 


451479 


451633 






283 


451786 


451940 


452093 


452247 


452400 


452553 


452706 


452859 


453012 


453165 






284 


453318 


453471 


453624 


453777 


453930 


454082 


454235 


454387 


454540 


454692 






285 


454845 


454997 


455150 


455302 


455454 


455606 


455758 


455910 


456062 


456214 






286 


456366 


456518 


456670 


456821 


456973 


457125 


457276 


457428 


457579 


457731 






287 


457882 


458033 


458184 


458336 


458487 


458638 


458789 


458940 


459091 


459242 






288 


459392 


459543 


459694 


459845 


459995 


460146 


460296 


460447 


460597 


460748 






289 


460898 


461048 


461198 


461348 


461499 


461649 


461799 


461948 


462098 


462248 






290 


462398 


462548 


462697 


462847 


462997 


463146 


463296 


463445 


463594 


463744 






291 


463893 


464042 


464191 


464340 


464490 


464639 


464788 


464936 


465085 


465234 






292 


465383 


465532 


465680 


465829 


465977 


466126 


466274 


466423 


466571 


466719 






293 


466868 


467016 


467164 


467312 


467460 


467608 


467756 


467904 


468052 


468200 






294 


468347 


468495 


468643 


468790 


468938 


469085 


469233 


469380 


469527 


469675 






295 


469822 


469969 


47011G 


470263 


470410 


470557 


470704 


470851 


470998 


471145 






296 


471292 


471438 


471585 


471732 


471878 


472025 


472171 


472318 


472464 


472610 






297 


472756 


472903 


473049 


473195 


473341 


473487 


473633 


473779 


473925 


474071 






298 


474216 


474362 


474508 


474653 


474799 


474944 


475090 


475235 


475381 


475526 


i 




299 


475671 


475816 


475962 


476107 


476252 


476397 


476542 


476687 


476832 


476976 






300 


477121 


477266 


477411 


477555 


477700 


477844 


477989 


478133 


478278 


478422 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



' 446 






ROGERS' DRAWING 


AND DESIGN. 
















TABLE OF 


LOGARITHMS-Continoed. 


m 


■ 








N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




301 


478566 


478711 


478855 


478999 


479143 


479287 


479431 


479575 


479719 


479863 




302 


480007 


480151 


480294 


480438 


480582 


480725 


480869 


481012 


481156 


481299 






303 


481443 


481586 


481729 


481872 


482016 


482159 


482302 


482445 


482588 


482731 






304 


482874 


483016 


483159 


483302 


483445 


483587 


483730 


483872 


484015 


484157 






305 


484300 


484442 


484585 


484727 


484869 


485011 


485153 


485295 


485437 


485579 






306 


485721 


485863 . 


486005 


486147 


486289 


486430 


486572 


486714 


486855 


486997 






307 


487138 


487280 


487421 


487563 


487704 


487845 


487986 


488127 


488269 


488410 






308 


488551 


488692 


488833 


488974 


489114 


489255 


489396 


489537 


489677 


489818 






309 


489958 


490099 


490239 


490380 


490520 


490661 


490801 


490941 


491081 


491222 






310 


491362 


491502 


491642 


491782 


491922 


492062 


492201 


492341 


492481 


492621 






311 


492760 


492900 


493040 


493179 


493319 


493458 


493597 


493737 


493876 


494015 






312 


494155 


494294 


494433 


494572 


494711 


494850 


494989 


495128 


495267 


495406 






313 


495544 


495683 


495822 


495960 


496099 


496238 


496376 


486515 


496653 


496791 






314 


496930 


497068 


497206 


497344 


497483 


497621 


497759 


497897 


498035 


498173 






315 


498311 


498448 


498586 


498724 


498862 


498999 


499137 


499275 


499412 


499550 






316 


499687 


499824 


499962 


500099 


500236 


500374 


500511 


500648 


500785 


500922 






317 


501059 


501196 


501333 


501470 


501607 


501744 


501880 


502017 


502154 


502291 






318 


502427 


502564 


502700 


502837 


502973 


503109 


503246 


503382 


503518 


503655 






319 


503791 


503927 


504063 


504199 


504335 


504471 


504607 


504743 


504878 


505014 






320 


505150 


505286 


505421 


505557 


505693 


505828 


505964 


506099 


506234 


506370 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



































ROGERS' DRAWING 


AND DESIGN. 






447 










TABLE OF 


LOGARITHMS- Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




321 


506505 


506640 


506776 


506911 


507046 


507181 


507316 


507451 


507586 


507721 




322 


507856 


507991 


508126 


508260 


508395 


508530 


508664 


508799 


508934 


509068 






323 


509203 


509337 


509471 


509606 


509740 


509874 


510009 


510143 


510277 


510411 






324 


510545 


510679 


510813 


510947 


511081 


511215 


511349 


511482 


511616 


511750 






325 


511883 


512017 


512151 


512284 


512418 


512551 


512684 


512818 


512951 


513084 






326 


513218 


513351 


513484 


513617 


513750 


513883 


514016 


514149 


514282 


514415 






327 


514548 


514681 


514813 


514946 


515079 


515211 


515344 


515476 


515609 


515741 






328 


515874 


516006 


516139 


516271 


516403 


516535 


516668 


516800 


516932 


517064 






329 


517196 


517328 


517460 


517592 


517724 


517855 


517987 


518119 


518251 


518382 






330 


518514 


518646 


518777 


518909 


519040 


519171 


519303 


519434 


519566 


519697 






331 


519828 


519959 


520090 


520221 


520353 


520484 


520615 


520745 


520876 


521007 






332 


521138 


521269 


521400 


521530 


521661 


521792 


521922 


522053 


522183 


522314 






333 


522444 


522575 


522705 


522835 


522966 


523096 


523226 


523356 


523486 


523616 






334 


523746 


523876 


524006 


524136 


524266 


524396 


524526 


524656 


524785 


524915 






335 


525045 


525174 


525304 


525434 


525563 


525693 


525822 


525951 


526081 


526210 






336 


526339 


526469 


526598 


526727 


526856 


526985 


527114 


527243 


527372 


527501 






337 


527630 


527759 


527888 


528016 


528145 


528274 


528402 


528531 


528660 


528788 






338 


528917 


529045 


529174 


529302 


529430 


529559 


529687 


529815 


529943 


530072 






339 


530200 


530328 


530456 


530584 


530712 


530840 


530968 


531096 


531223 


531351 






340 


531479 


531607 


531734 


531862 


531990 


532117 


532245 


532372 


532500 


532627 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 





























448 



ROGERS' DRAWING AND DESIGN. 









TABLE OF LOGARITHMS-Continoed. 








N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


341 


532754 


^532882 


533009 


533136 


533264 


533391 


533518 


533645 


533772 


533899 


342 


534026 


534153 


534280 


534407 


534534 


534661 


534787 


534914 


535041 


535167 


343 


535294 


535421 


535547 


535674 


535800 


535927 


536053 


536180 


536306 


536432 


344 


536558 


536685 


536811 


536937 


537063 


537189 


537315 


537441 


537567 


537693 


346 


537819 


537945 


538071 


538197 


538822 


538448 


538574 


538699 


538825 


538951 


346 


539076 


539202 


539327 


539452 


539578 


539703 


539829 


539954 


540079 


540204 


347 


540329 


540455 


540580 


540705 


540830 


540955 


541080 


541205 


541330 


541454 


348 


541579 


541704 


541829 


541953 


542078 


542203 


542327 


542452 


542576 


542701 


349 


542825 


542950 


543074 


543199 


543323 


543447 


543571 


543696 


543820 


543944 


350 


544068 


544192 


544316 


544440 


544564 


544688 


544812 


544936 


545060 


545183 


351 


545307 


545431 


545555 


545678 


545802 


545925 


546049 


546172 


546296 


546419 


352 


546543 


546666 


546789 


546913 


547036 


547159 


547282 


547405 


547529 


547652 


353 


547775 


547898 


548021 


548144 


548267 


548389 


548512 


548635 


548758 


548881 


354 


549003 


549126 


549249 


549371 


549494 


549616 


549739 


549861 


549984 


550106 


355 


550228 


550351 


550473 


550595 


550717 


550840 


550962 


551084 


551206 


551328 


356 


551450 


551572 


551694 


551816 


551938 


552060 


552181 


552303 


552425 


552547 


357 


552668 


552790 


552911 


553033 


553155 


553276 


553398 


553519 


553640 


553762 


358 


553883 


554004 


554126 


554247 


554368 


554489 


554610 


554731 


554852 


554973 


359 


555094 


555215 


555336 


555457 


555578 


555699 


555820 


555940 


556061 


556182 


360 


556303 


556423 


556544 


556664 


556785 


556905 


557026 


557146 


557267 


557387 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



ROGERS' DRAWING AND DESIGN. 



449 









TABLE OF ] 


LOGARITHMS-Continued. 








N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


361 


557507 


557627 


557748 


557868 


557988 


558108 


558228 


558349 


558469 


558589 


362 


558709 


558829 


558948 


559068 


559188 


559308 


559428 


559548 


559667 


559787 


363 


559907 


560026 


560146 


560265 


560385 


560504 


560624 


560743 


560863 


560982 


364 


561101 


561221 


561340 


561459 


561578 


561698 


561817 


561936 


562055 


562174 


365 


562293 


562412 


562531 


562650 


562769 


562887 


563006 


563125 


563244 


563362 


366 


563481 


563600 


563718 


563837 


563955 


564074 


564192 


564311 


564429 


564548 


367 


564666 


564784 


564903 


565021 


565139 


565257 


565376 


565494 


565612 


565730 


368 


565848 


565966 


566084 


566202 


566320 


566437 


566555 


566673 


566791 


566909 


369 


567026 


567144 


567262 


567379 


567497 


567614 


567732 


567849 


567967 


568084 


370 


568202 


568319 


568436 


568554 


568671 


568788 


56890a 


569023 


569140 


569257 


371 


569374 


569491 


569608 


569725 


569842 


569959 


570076 


570193 


570309 


570426 


372 


570543 


570660 


570776 


570893 


571010 


571126 


571243 


571359 


571476 


571592 


373 


571709 


571825 


571942 


572058 


572174 


572291 


572407 


572523 


572639 


572755 


374 


572872 


572988 


573104 


573220 


573336 


573452 


573568 


573684 


573800 


573915 


375 


574031 


574147 


574263 


574379 


574494 


574610 


574726 


574841 


574957 


575072 


376 


575188 


575303 


575419 


575534 


575650 


575765 


575880 


575996 


576111 


576226 


377 


576341 


576457 


576572 


576687 


576802 


576917 


577032 


577147 


577262 


577377 


378 


577492 


577607 


577722 


577836 


577951 


578066 


578181 


578295 


578410 


578525 


379 


578639 


578754 


578868 


578983 


579097 


579212 


579326 


579441 


579555 


579669 


380 


579784 


579898 


580012 


580126 


580241 


580355 


580469 


580583 


580697 


580811 


N 





1 


2 


3 


4 


6 


6 


7 


8 


9 



450 






ROGERS' DRAWING 


AND DESIGN 
















TABLE OF ] 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




381 


580925 


581039 


581153 


581267 


581381 


581495 


581608 


581722 


581836 


581950 




382 


582063 


582177 


582291 


582404 


582518 


582631 


582745 


582858 


582972 


583085 






383 


583199 


583312 


583426 


583539 


583652 


583765 


583879 


583992 


584105 


584218 






384 


584331 


584444 


584557 


584670 


584788 


584896 


585009 


585122 


585235 


585348 






385 


585461 


585574 


585686 


585799 


685912 


586024 


586137 


586250 


586362 


586475 






386 


586587 


586700 


586812 


586925 


587037 


587149 


587262 


587374 


587486 


587599 






387 


587711 


587823 


587935 


588047 


588160 


588272 


588384 


588496 


588608 


588720 






388 


588832 


588944 


589056 


589167 


589279 


589391 


589503 


589G15 


589726 


589838 






389 


589950 


590061 


590173 


590284 


590396 


590507 


690619 


590730 


590842 


590953 






390 


591065 


591176 


591287 


591399 


591510 


591621 


591732 


591843 


591955 


592066 






391 


592177 


592288 


592399 


592510 


592621 


592732 


592843 


592954 


593064 


593175 






392 


593286 


593397 


593508 


593618 


593729 


593840 


593950 


594061 


594171 


594282 






393 


594393 


594503 


594614 


594724 


594834 


594945 


595065 


595165 


595276 


595386 






394 


595496 


595606 


595717 


595827 


595937 


596047 


596157 


596267 


596377 


596487 






395 


596597 


596707 


596817 


596927 


597037 


597146 


597256 


597366 


597476 


597586 






396 


597695 


597805 


597914 


598024 


598134 


598243 


598353 


598462 


598572 


598681 






397 


598791 


598900 


599009 


599119 


599228 


599337 


599446 


599556 


599665 


599774 






398 


599883 


599992 


600101 


600210 


600319 


600428 


600537 


600646 


600755 


600864 






399 


600973 


601082 


601191 


601299 


601408 


601517 


601625 


601734 


601843 


601951 






400 


602060 


602169 


602277 


602386 


602494 


602603 


G02711 


602819 


602928 


603036 




N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



































ROGERS' DRAWING 


AND DESIGN. 






451 










TABTF, OF ] 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




401 


603144 


603253 


603361 


603469 


603577 


603686 


603794 


603902 


604010 


604118 




402 


604226 


604334 


604442 


604550 


604658 


604766 


604874 


604982 


605089 


605197 






403 


605305 


605413 


605521 


605628 


605736 


605844 


605951 


606059 


606166 


606274 






404 


606381 


606489 


606596 


606704 


606811 


606919 


607026 


607133 


607241 


607348 






405 


607455 


607562 


607669 


607777 


607884 


607991 


608098 


608205 


608312 


608419 






406 


608526 


608633 


608740 


608847 


608954 


609061 


609167 


609274 


609381 


609488 






407 


609594 


609701 


609808 


609914 


610021 


610128 


610234 


610341 


610447 


610554 






408 


610660 


610767 


610873 


610979 


611086 


611192 


611298 


611405 


611511 


611617 






409 


611723 


611829 


^ 611936 


612042 


612148 


612254 


612360 


612466 


612572 


612678 






410 


612784 


612890 


612996 


613102 


613207 


613313 


613419 


613525 


613630 


613736 






411 


613842 


613947 


614053 


614159 


614264 


614370 


614475 


614581 


614686 


614792 






412 


614897 


615003 


615108 


615213 


615319 


615424 


615529 


615634 


615740 


615845 






413 


615950 


616055 


616160 


616265 


616370 


616476 


616581 


616686 


616790 


616895 






414 


617000 


617105 


617210 


617315 


617420 


617525 


617629 


617734 


617839 


617948 






416 


618048 


618153 


618257 


618362 


618466 


618571 


618676 


618780 


618884 


618989 






416 


619093 


619198 


619302 


619406 


619511 


619615 


619719 


619824 


619928 


620032 






417 


620136 


620240 


620344 


620448 


620552 


620656 


620760 


620864 


620968 


621072 






418 


621176 


621280 


621384 


621488 


621592 


621695 


621799 


621903 


622007 


622110 






419 


622214 


622318 


622421 


622525 


622628 


622732 


622835 


622939 


623042 


623146 






420 


623249 


623353 


623456 


623559 


623663 


623766 


623869 


623973 


624076 


624179 




N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


























1 



452 



ROGERS' DRAWING AND DESIGN. 









TABLE OF ] 


LOGARITHMS- Com 


4nued. 








N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


421 


624282 


624385 


624488 


624591 


624695 


624798 


624901 


625004 


625107 


625210 


422 


625312 


625415 


625518 


625621 


625724 


625827 


625929 


626032 


626135 


626238 


423 


626340 


626443 


626546 


626648 


626751 


626853 


626956 


627058 


627161 


627263 


424 


627366 


627468 


627571 


627673 


627775 


627878 


627980 


628082 


628185 


628287 


425 


628389 


628491 


628593 


628695 


628797 


628900 


629002 


629104 


629206 


629308 


426 


629410 


629512 


629613 


629715 


629817 


629919 


630021 


630123 


630224 


630326 


427 


630428 


630530 


630631 


630733 


630835 


630936 


631038 


631139 


631241 


631342 


428 


631444 


631545 


631647 


631748 


631849 


631951 


632052 


632153 


632255 


632356 


429 


632457 


632559 


632660 


632761 


632862 


632963 


633064 


633165 


633266 


633367 


430 


633468 


633569 


633670 


633771 


633872 


633973 


634074 


634175 


634276 


634376 


431 


634477 


634578 


634679 


634779 


634880 


634981 


635081 


635182 


635283 


635383 


432 


635484 


635584 


635685 


635785 


635886 


635986 


636087 


636187 


636287 


636388 


433 


636488 


636588 


636688 


636789 


636889 


636989 


637089 


637189 


637290 


637390 


434 


637490 


637590 


637690 


637790 


637890 


637990 


638090 


638190 


638290 


638389 


435 


638489 


638589 


638689 


638789 


638888 


638988 


639088 


639188 


639287 


639387 


436 


639486 


639586 


639686 


639785 


639885 


639984 


640084 


640183 


640283 


640382 


437 


640481 


640581 


640680 


640779 


640879 


640978 


641077 


641177 


641276 


641375 


438 


641474 


641573 


641G72 


641771 


641871 


641970 


642069 


642168 


642267 


642366 


439 


642465 


642563 


642662 


G42761 


642860 


642959 


643058 


643156 


643255 


643354 


440 


643453 


643551 


643650 


643749 


643847 


643946 


644044 


644143 


644242 


644340 


N 





1 


2 


2 


4 


5 


6 


7 


8 


9 









ROGERS' DRAWING 


AND DESIGN. 






453 










TABLE OF 


LOGARITHMS— Continued 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




441 


644439 


644537 


644636 


644734 


644832 


644931 


645029 


645127 


645226 


645324 




442 


645422 


645521 


645619 


645717 


645815 


645913 


646011 


646110 


646208 


646306 






443 


646404 


646502 


646600 


646698 


646796 


646894 


646992 


647089 


647187 


647285 






444 


647383 


647481 


647579 


647676 


647774 


647872 


647969 


6480G7 


648165 


648262 






445 


648360 


648458 


648555 


648653 


648750 


648848 


648945 


649043 


649140 


649237 






446 


649335 


649432 


649530 


649627 


649724 


649821 


649919 


650016 


650113 


650210 






447 


650308 


650405 


650502 


650599 


650696 


650793 


650890 


650987 


651084 


651181 






448 


651278 


651375 


651472 


651569 


651666 


651762 


651859 


651956 


652053 


652150 






449 


652246 


652343 


652440 


652536 


652633 


652730 


652826 


652923 


653019 


653116 






460 


653213 


653309 


653405 


653502 


653598 


653695 


653791 


653888 


653984 


654080 






451 


654177 


654273 


654369 


654465 


654562 


654658 


654754 


654850 


654946 


655042 






452 


655138 


655235 


655331 


655427 


655523 


655619 


655715 


655810 


655906 


656002 






463 


656098 


656194 


656290 


656386 


656482 


656577 


656673 


656769 


656864 


656960 






454 


657056 


657152 


657247 


657343 


657438 


657534 


657629 


657725 


657820 


657916 






456 


658011 


658107 . 


658202 


658298 


658393 


658488 


658584 


658679 


658774 


658870 






466 


658965 


659060 


659155 


659250 


659346 


659441 


659536 


659631 


659726 


659821 






457 


659916 


660011 


660106 


660201 


660296 


660391 


660486 


660581 


660676 


660771 






458 


660865 


660960 


661055 


661150 


661245 


661339 


661434 


661529 


661623 


661718 






459 


661813 


661907 


662002 


662096 


662191 


662286 


662380 


662475 


662569 


662663 






460 


662758 


662852 


662947 


663041 


663135 


663230 


663324 


663418 


663512 


663607 




N 





1 


2 


3 


i 


6 


6 


7 


8 


9 





























454 



ROGERS' DRAWING AND DESIGN. 



TABLE OF LOGAPaTHMS— Continued. 



IT 





1 


& 


3 


4 


5 


6 


7 


8 


9 


461 


663701 


663795 


663889 


663983 


664078 


664172 


664266 


664360 


664454 


664548 


462 


664642 


664736 


664830 


664924 


665018 


665112 


665206 


665299 


665393 


665487 


463 


665581 


665675 


665769 


665862 


665956 


666050 


666143 


666237 


666331 


666424 


464 


666518 


666612 


666705 


666799 


666892 


666986 


667079 


667173 


667266 


667360 


465 


667453 


667546 


667640 


667733 


667826 


667920 


668013 


668106 


668199 


668293 


4^6 


668386 


668479 


668572 


668665 


668759 


668852 


1 668945 


669038 


669131 


669224 


467 


669317 


669410 


669503 


669596 


669689 


669782 


669875 


669967 


670060 


670153 


468 


670246 


670339 


670431 


670524 


670617 


670710 


670802 


670895 


670988 


671080 


469 


671173 


671265 


671358 


671451 


671543 


671636 


671728 


671821 


671913 


672005 


470 


672098 


672190 


672283 


672375 


672467 


672560 


672662 


672744 


672836 


672929 


471 


673021 


673113 


673205 


673297 


673390 


673482 


673574 


673666 


673758 


673850 


472 


673942 


674034 


674126 


674218 


674310 


674402 


674494 


674586 


674677 


G74769 


473 


674861 


674953 


675045 


675137 


675228 


675320 


675412 


675503 


675595 


675687 


474 


675778 


675870 


675962 


676053 


676145 


676236 


676328 


676419 


676511 


676602 


475 


676694 


676785 


676876 


676968 


677059 


677151 


677242 


677333 


677424 


677516 


476 


677607 


677698 


677789 


677881 


677972 


678063 


678154 


678245 


678336 


678427 


477 


678518 


678609 


678700 


678791 


678882 


678973 


679064 


679155 


679246 


679337 


478 


679428 


679519 


679610 


679700 


679791 


679882 


679973 


680063 


680154 


680245 


479 


680336 


680426 


680517 


680607 


680698 


680789 


680879 


680970 


681060 


681151 


480 


681241 


681332 


681422 


681513 


681603 


681693 


681784 


681874 


681964 


682055 


N 





1 


2 


3 


4 


5 


6 


7 


8^ 


9 



ROGERS' DRAWING AND DESIGN. 



455 









TABLE OF ] 


LOGARITHMS- Continued. 








N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


481 


682145 


682235 


682326 


682416 


682506 


682596 


682686 


682777 


682867 


682957 


482 


683047 


683137 


683227 


683317 


683407 


683497 


683587 


683677 


683767 


683857 


483 


683947 


684037 


684127 


684217 


684307 


684396 


684486 


684576 


684666 


684756 


484 


684845 


684935 


685025 


685114 


685204 


685294 


685383 


685473 


685563 


685652 


485 


685742 


685831 


685921 


686010 


686100 


686189 


686279 


686368 


686458 


686547 • 


486 


686636 


686726 


686815 


686904 


686994 


687083 


687172 


687261 


687351 


687440 


487 


687529 


687618 


687707 


687796 


687886 


687975 


688064 


688153 


688242 


688331 


488 


688420 


688509 


688598 


688687 


688776 


688865 


688953 


689042 


689131 


689220 


489 


689309 


689398 


689486 


689575 


689664 


689753 


689841 


689930 


690019 


690107 


490 


690196 


690285 


690373 


690462 


690550 


690639 


690728 


690816 


690905 


690993 


491 


691081 


6911-70 


691258 


691347 


691435 


691524 


691612 


691700 


691789 


691877 


492 


691965 


692053 


692142 


692230 


692318 


692406 


692494 


692583 


692671 


692759 


493 


692847 


692935 


693023 


693111 


693199 


693287 


693375 


693463 


693551 


693639 


494 


693727 


693815 


693903 


693991 


694078 


694166 


694254 


694342 


694430 


694517 


495 


694605 


694693 


694781 


694868 


694956 


695044 


695131 


695219 


695307 


695394 


496 


695482 


695569 


695657 


695744 


695832 


695919 


696007 


696094 


696182 


696269 


497 


696356 


696444 


696531 


696618 


696706 


696793 


696880 


696968 


697055 


697142 


498 


697229 


697317 


697404 


697491 


697578 


697665 


697752 


697839 


697926 


698014 


499 


698101 


698188 


698275 


698362 


698449 


698535 


698622 


698709 


698796 


698883 


500 


698970 


699057 


699144 


699231 


699317 


699404 


699491 


699578 


699664 


699751 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



456 



ROGERS' DRAWING AND DESIGN. 



TABLE OF LOGARITHMS— Continued. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


501 


699838 


699924 


700011 


700098 


700184 


700271 


700358 


700444 


700531 


700617 


502 


700704 


700790 


700877 


700963 


701050 


701136 


701222 


701309 


701395 


701482 


503 


701568 


701654 


701741 


701827 


701913 


701999 


702086 


702172 


702258 


702344 


504 


702431 


702517 


702603 


702689 


702775 


702861 


702947 


703033 


703119 


703205 


505 


703291 


703377 


703463 


703549 


703635 


703721 


703807 


703893 


703979 


704065 


506 


704151 


704236 


704322 


704408 


704494 


704579 


704665 


704751 


704837 


704922 


507 


705008 


705094 


705179 


705265 


705350 


705436 


705522 


705607 


705693 


705778 


508 


705864 


705949 


706035 


706120 


706206 


706291 


706376 


706462 


706547 


706632 


509 


706718 


706803 


706888 


706974 


707059 


707144 


707229 


707315 


707400 


707485 


510 


707570 


707655 


707740 


707826 


707911 


707996 


708081 


708166 


708251 


708336 


511 


708421 


708506 


708591 


708676 


708761 


708846 


708931 


709015 


709100 


709185 


512 


709270 


709355 


709440 


709524 


709609 


709694 


709779 


709863 


709948 


710033 


513 


710117 


710202 


710287 


710371 


710456 


710540 


710625 


710710 


710794 


710879 


514 


710963 


711048 


711132 


711217 


711301 


711385 


711470 


711554 


711639 


711723 


515 


711807 


711892 


711976 


712060 


712144 


712229 


712313 


712397 


712481 


712566 


516 


712650 


712734 


712818 


712902 


712986 


713070 


713154 


713238 


713323 


713407 


517 


713491 


713575 


713659 


713742 


713826 


713910 


713994 


714078 


714162 


714246 


518 


714330 


714414 


714497 


714581 


714665 


714749 


714833 


714916 


715000 


715084 


519 


715167 


715251 


715335 


715418 


715502 


715586 


715669 


715753 


715836 


715920 


520 


716003 


716087 


716170 


716254 


716337 


716421 


716504 


716588 


716671 


716754 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



ROGERS' DRAWING AND DESIGN. 



457 



TABLE OF LOGARITHMS— Continued. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


521 


716838 


716921 


717004 


717088 


717171 


717254 


717338 


717421 


717504 


717587 


522 


717671 


717754 


717837 


717920 


718003 


718086 


718169 


718253 


718336 


718419 


523 


718502 


718585 


718668 


718751 


718834 


718917 


719000 


719083 


719165 


719248 


524 


719331 


719414 


719497 


719580 


719663 


719745 


719828 


719911 


719994 


720077 


625 


720159 


720242 


720325 


720407 


720490 


720573 


720655 


720738 


720821 


720903 


526 


720986 


721068 


721151 


721233 


721316 


721398 


721481 


721563 


721646 


721728 


527 


721811 


721893 


721975 


722058 


722140 


722222 


722305 


722387 


722469 


722552 


528 


722634 


722716 


722798 


722881 


722963 


723045 


723127 


723209 


723291 


723374 


529 


723456 


723538 


723620 


723702 


723784 


723866 


723948 


724030 


724112 


724194 


530 


724276 


724358 


724440 


724522 


724604 


724685 


724767 


724849 


724931 


725013 


531 


725095 


725176 


725258 


725340 


725422 


725503 


725585 


725667 


725748 


725830 


632 


725912 


725993 


726075 


726156 


726238 


726320 


726401 


726483 


726564 


726646 


533 


726727 


726809 


726890 


726972 


727053 


727134 


727216 


727297 


727379 


727460 


534 


727541 


727623 


727704 


727785 


727866 


727948 


728029 


728110 


728191 


728273 


535 


728354 


728435 


728516 


728597 


728678 


728759 


728841 


728922 


729003 


729084 


536 


729165 


729246 


729327 


729408 


729489 


729570 


729651 


729732 


729813 


729893 


537 


729974 


730055 


730136 


730217 


730298 


730378 


730459 


730540 


730621 


730702 


538 


730782 


730863 


730944 


731024 


731105 


731186 


731266 


731347 


731428 


731508 


539 


731589 


731669 


731750 


731830 


731911 


731991 


732072 


732152 


732233 


732313 


540 


732394 


732474 


732555 


732635 


732715 


732796 


732876 


732956 


733037 


733117 


N 





1 


2 


3 


4 


6 


6 


7 


8 


9 



458 



ROGERS' DRAWING AND DESIGN. 









TABLE OF LOGARITHMS— Continued. 








N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


641 


733197 


733278 


733358 


733438 


733518 


733598 


733679 


733759 


733839 


733919 


542 


733999 


734079 


734160 


734240 


734320 


734400 


734480 


734560 


734640 


734720 


543 


734800 


734880 


734960 


735040 


735120 


735200 


735279 


735359 


735439 


735519 


544 


735599 


735679 


735759 


735838 


735918 


735998 


736078 


736157 


736237 


736317 


645 


736397 


736476 


736556 


736635 


736715 


736795 


736874 


736954 


737034 


737113 


546 


737193 


737272 


737352 


737431 


737511 


737590 


737670 


737749 


737829 


737908 


647 


737987 


738067 


738146 


738225 


738305 


738384 


738463 


738543 


738622 


738701 


548 


738781 


738860 


738939 


739018 


739097 


739177 


739256 


739335 


739414 


739493 


549 


739572 


739651 


739731 


739810 


739889 


739968 


740047 


740126 


740205 


740284 


550 


740363 


740442 


740521 


740600 


740678 


740757 


740836 


740915 


740994 


741073 


551 


741152 


741230 


741309 


741388 


741467 


741546 


741624 


741703 


741782 


741860 


552 


741939 


742018 


742096 


742175 


742254 


742332 


742411 


742489 


742568 


742647 


553 


742725 


742804 


742882 


742961 


743039 


743118 


743196 


743275 


743353 


743431 


554 


743510 


743588 


743667 


743745 


743823 


743902 


743980 


744058 


744136 


744215 


555 


744293 


744371 


744449 


744528 


744606 


744684 


744762 


744840 


744919 


744997 


556 


745075 


745153 


745231 


745309 


745387 


745465 


745543 


745621 


745699 


745777 


557 


745855 


745933 


746011 


746089 


746167 


746245 


746323 


746401 


746479 


746556 


558 


746634 


746712 


746790 


746868 


746945 


747023 


747101 


747179 


747256 


747334 


559 


747412 


747489 


747567 


747645 


747722 


747800 


747878 


747955 


748033 


748110 


660 


748188 


748266 


748343 


748421 


748498 


748576 


748653 


748731 


748808 


748885 


N 





~ - 

1 


2 


3 


4 


5 


6 


7 


8 


9 



ROGERS' DRAWING AND DESIGN. 



459 



TABLE OF LOGARITHMS— Continued. 



N 





1 


2 


3 


4 


5 


— 

6 


7 


8 


9 


561 


74S9G3 


749040 


749118 


749195 


749272 


749350 


749427 


749504 


749582 


749659 


562 


749736 


749814 


749891 


749968 


750045 


750123 


750200 


750277 


750354 


750431 


563 


750508 


750586 


750663 


750740 


750817 


750894 


750971 


751048 


751125 


751202 


564 


751279 


751356 


751433 


751510 


751587 


751664 


751741 


751818 


751895 


751972 


565 


75204h 


752125 


752202 


752279 


752356 


752433 


752509 


752586 


752663 


752740 


566 


752816 


752893 


752970 


753047 


753123 


753200 


753277 


753353 


753430 


753506 


567 


753583 


753660 


753736 


753813 


753889 


753966 


754042 


754119 


754195 


754272 


568 


754348 


754425 


754501 


754578 


754654 


754730 


754807 


754883 


754960 


755036 


569 


755112 


755189 


755265 


755341 


755417 


755494 


755570 


755646 


755722 


755799 


570 


755875 


755951 


756027 


756103 


756180 


756256 


756332 


756408 


756484 


756560 


571 


756636 


756712 


756788 


756864 


756940 


757016 


757092 


757168 


757244 


757320 


572 


757396 


757472 


757548 


757624 


757700 


757775 


757851 


757927 


758003 


758079 


673 


758155 


758230 


758306 


758382 


758458 


758533 


758609 


758685 


758761 


758836 


574 


758912 


758988 


759063 


759139 


759214 


759290 


759366 


759441 


759517 


759592 


57-5 


759668 


759743 


759819 


759894 


759970 


760045 


760121 


760196 


760272 


760347 


576 


760422 


760498 


760573 


760649 


760724 


760799 


760875 


760950 


761025 


761101 


577 


761176 


761251 


761326 


761402 


761477 


761552 


761627 


761702 


761778 


761853 


578 


761928 


762003 


762078 


762153 


762228 


762303 


762378 


762453 


762529 


762604 


579 


762679 


762754 


762829 


762904 


762978 


763053 


763128 


763203 


763278 


763353 


680 


763428 


763503 


763578 


763653 


763727 


763802 


763877 


763952 


764027 


764101 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



460 






ROGERS' DRAWING 


AND DESIGN. 






- 










TABLE OF 


LOGARITHMS— Continued. 












N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




681 


764176 


764251 


764326 


764400 


764475 


764550 


764624 


764699 


764774 


764848 




582 


764923 


764998 


765072 


765147 


765221 


765296 


765370 


765445 


765520 


765594 






683 


765669 


765743 


765818 


765892 


765966 


766041 


766115 


766190 


766264 


766338 






684 


766413 


766487 


766562 


766636 


766710 


766785 


766859 


766933 


767007 


767082 






686 


767156 


767230 


767304 


767379 


767453 


767527 


767601 


767675 


767749 


767823 






686 


767898 


767972 


768046 


768120 


768194 


768268 


768342 


768416 


768490 


768564 






687 


768638 


768712 


768786 


768860 


768934 


769008 


769082 


769156 


769230 


769303 






688 


769377 


769451 


769525 


769599 


769673 


769746 


769820 


769894 


769968 


770042 






589 


770115 


770189 


770263 


770336 


770410 


770484 


770557 


770631 


770705 


770778 






690 


770852 


770926 


770999 


771073 


771146 


771220 


771293 


771367 


771440 


771514 






591 


771587 


771661 


771734 


771808 


771881 


771955 


772028 


772102 


772175 


772248 






692 


772322 


772395 


772468 


772542 


772615 


772688 


772762 


772835 


772908 


772981 






693 


773055 


773128 


773201 


773274 


773348 


773421 


773494 


773567 


773640 


773713 






594 


773786 


773860 


773933 


774006 


774079 


774152 


774225 


774298 


774371 


774444 






595 


774517 


774590 


774663 


774736 


774809 


774882 


774955 


775028 


775100 


775173 






596 


775246 


775319 


775392 


775465 


775538 


775610 


775683 


775756 


775829 


775902 


' 




597 


775974 


776047 


776120 


776193 


776265 


776338 


776411 


776483 


776556 


776629 






598 


776701 


776774 


776846 


776919 


776992 


777064 


777137 


777209 


777282 


777354 


1 




599 


777427 


777499 


777572 


777644 


777717 


777789 


777862 


777934 


778006 


778079 


1 




600 


778151 


778224 


778296 


778368 


778441 


778513 


778585 


778658 


778730 


778802 


1 
1 


N 





1 


2 


3 


4 


6 


6 


7 


9 


9 














































J 



USEFUL TABLES FOR DRAUGHTSMEN, MACHINISTS 




AND ENGINEERS. 






TABLE OF DECIMAL EQUIVALENTS. 






Sths, J6ths, 32ds and 64tlis of an Inch. 




8ths. 


32nds. 


64tbs. 


Il=-5i5625 


J =.125 


A= -03125 


,V= -015625 


lt= 


-546875 


i-.25o 


^=•09375 


5\=. 046875 


ff= 


578125 


f =-375 


^=•15625 


5\ = .o78i25 


11= 


609375 


^=.500 


A= .21875 


A=- 109375 


ii= 


640625 


1= 625 


A= -28125 


^=.140625 


H= 


.671875 


f =-75o 


M= -34375 


I4=.i7i875 


n= 


703125 


l=.875 


J|=. 40625 


|f=.203I25 


u= 


734375 


i6ths. 


M=-46875 


wX. 234375 


11= 


765625 


f J =.0625 


Ji = -53i25 


H= -265625 


u--= 


796875 


A=-i875 


M=-59375 


^ = .296875 


11= 


828125 


A=-3I25 


M= -65625 


li=.32Si25 


11= 


859375 


TJ=-4375 


11= -71875 


ll=-359375 


u= 


S90625 


^=•5625 


If =-78125 


11= .390625 


M= 


921S75 


ii = .6875 


fl=-84375 


||=.42i875 


U= 


953125 


n=.8i25 


||=.9o625 


||=.453i25 


Sf= 


984375 


11= -9375 


M=-96875 


|i=. 484375 







461 





d62 










ROGERS' DRAWING AND DESIGN. 








1— 1 
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1— 1 

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Q 

00 


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1 

O 
H 


1 

a 

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a 

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1 

W 

o 

V3 

G 

-i 

a 
S 

a 


JO joqiun^ 


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8°° 










I© Ift »0 5^ 

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^-r-*CO?CCO(M'^4'MCN(M'M"r--.— r^t-H,— OOOOOOOOOOOOOOOOOOOOOOOOOO • • 








•^J-Hi-HOOOCiC50a500QOOOCOIr*t^l>'CDCDO»0»Oir5U*i'^'^'-i'^7COOirNt--'-i— 'ooooasoi 

• . • .c^(^^(^l(r^(^^c^l^r-^l-HI-HrH^HIHr-f-H.-HrHrM,-^rHl--^T-HI-Hr^,-HrHr-r^r^rHT-l^-HrH,^ 






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-fOSCD-ri^OCOOfM-^COOOOCqOO 

^•^TfCOCCCOC^iMC^OJTJi-'i— — 1— .— f-^ocooooooooooooooooooooooooo<:> 






■SM'Jaie33J0A\ 
'■00 -ajjvr naow 
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. .00»OOiOO»OI>.CCOOOOCOOiO»ftlCOC>lOOiOOOOl^COOOO-^^fO(NOO'MOOOO"-HlC50 - - - » 

• .coc<I^ococ^col0^^c^^^'rlootoOlOT— 0'rio4^r^^-'^r-.ootncoocoi--cr'in'rfcocMr-*oooi - - - - 

• .Ciocoococc>"^<^^oo^^o-TCcc^oocol>.o»0"^'^coro<^^'^^'^^':^l.^-^I— c^rH^^r-*,— «r-ioo - - - - 

• •CC00C0«<M0*M(MC1^^i-'i-(^^^OOOOOOOOOOOOOOOOOOOOOOOO - - - - 






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JO 

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■ •tO'NOO'^ COiOCOiMOOOC^^tC'MOCiajl^OtO'Tt'-^COC^'M'ri'rJ'M.-ii-ir-i-lr-T-HOOOOO • - • - 

■ •'^'^COCOCCC^(MCN<?lG^^^^ — ^•-'OOOOOOOOOOOOOOOOOOOOOOOO • • - - 






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. ■ 0'*'^c:)i>.o-t' — r^^'^oO'^'--oOl— '"^r'-c^iOOKtii— i(>D<-:cioi>.»c-^c^i— 'Ocoi'-t^'iDioiO'^^ 






JO jaqmnii 







ROGERS' DRAWING AND DESIGN. 






463 




RULES 






















TABLES. 








Relative to the Qrcle. 


















Area, and Qrcomfei-enccs of Circles 


advancing^ by tenths. 




To And the area of a circle — 

Multiply circumferencebyone-quarterofthediameter. 
Or multiply the square of diameter by 0.7854. 
Or " '' circumference " .07958. 
















DIam. 


Area. 


CIrciun. 


Dloin. 


Area. 


Clrcum. 




0.0 

.1 


.007854 


.31416 


3.0 
.1 


7.0686 
7.5477 


9.4248 
9.7389 




Or " " ^ diameter " 3.1416. 


'.s 


.031416 
.070686 


.62832 
.94248 


.2 
.3 


8.0425 
8 5530 


10.0531 
10.3673 






A 


.12566 


1.2566 


.4 


9.0792 


10.6814 




To £nd circumference — 


.5 


.19735 


1.5708 


.5 


9.6211 


10.9956 




Multiply diameter by 3.1416. 


.6 

.7 


.28274 
.88485 


1.8850 
2.1991 


.6 

.7 


10.1788 
10.7521 


11.3097 
11.6239 




Or divide " " 0.3183. 


.8 
.9 


.50266 
.63617 


2.5133 
2.8274 


.8 
.9 


11.3411 
11.9456 


11.9381 
12.2523 




To And diameter — 


1.0 
.1 


.7854 
.9508 


3.1418 
8.4558 


4.0 
.1 


12.5664 
13.2025 


12.5664 
12.8805 




Multiply circumference by 0.3183. 


.2 
.3 


1.1810 
1.3273 


3.7699 
4.0841 


.2 
.3 


13.8544 
14.5220 


13.1947 
13.5088 




Or divide " " 3.1416. 


.4 


1.5394 


4,8982 


.4 


15.2053 


18.8230 






,5 


1.7671 


4.7124 


.5 


15.9048 


14.1372 




To And Radius — 


.6 


2.0106 


5.0265 


.6 


16.6190 


14.4513 






,7 


2.2698 


5.3407 


.7 


17.3494 


14.7655 




Multiply circumference by 0. ISQIS- 


.8 


2.5447 


5.6549 


.8 


18.0956 


15.0796 




Or divide " " 6.28318. 


.9 


2.8353 


5.9690 


.9 


18.8574 


15.8938 






2.0 


3.1416 


6.2832 


5.0 


19.6350 


15.7080 




In the following tables the diameter of a given 


.1 
.2 


8.4636 
3.8018 


6.5973 
6.9115 


.1 
.2 


20.4282 
21.2872 


16.0221 
16.3863 




inch is to be found in the first column, the area is to 


.3 


4.1548 


7.2257 


.8 


22.0618 


16.6504 






.4 


4.5239 


7.6398 


.4 


22.9023 


16.9646 




be found in the second column, and the circumference 
















in the third column. 


.5 
.6 


4.9087 
5.3093 


7.8540 
8.1681 


.5 
.6 


23.7588 
24.6301 


17.2788 
17.5929 




Example : A circle with a diameter of 2.7 inches 


.7 
.8 


5.7256 
6.1575 


8.4828 
8.7965 


.7 
.8 


25.5176 
26.4208 


17.9071 
18.2212 




has an area of 5.7256 square inches and a circumfer- 
ence of 8.4823 linear inches. 


.9 


6.6052 


9.1106 


.9 


37.8397 


18.5854 









464 



ROGERS' DRAWING AND DESIGN. 



TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. 



DIam. 


Area. 


Clrcum. 


Dlaiii. 


Area. 


CIrcuni. 


6.0 


28.8743 


18.8496 


10.0 


78.5398 


31.4159 


.1 


29.2247 


19.1637 


.1 


80.1185 


31.7301 


.2 


30.1907 


19.4779 


2 


81.7128 


32.0443 


.3 


31.1725 


19.7920 


.3 


83.3239 


33.3584 


.4 


32.1699 


20.1062 


.4 


84.9487 


32.6726 


.5 


33.1831 


20.4204 


.5 


86.5901 


32.9867 


.6 


34.8119 


20.7345 


.6 


88.3473 


33.3009 


.7 


35.2565 


21.0487 


.7 


89.9202 


33.6150 


.8 


36.3168 


21.3628 


.8 


91.6088 


33.9293 


.9 


37.3928 


21.6770 


.9 


93.3132 


34.2434 


7.0 


38.4845 


21.9911 


11.0 


95.0332 


34.5575 


.1 


39.5919 


28.3053 


.1 


96.7689 


34.8717 


.2 


40.7150 


82.6195 


.2 


98.'5203 


35.1858 


.3 


41.8539 


22.9336 


.3 


100.2875 


35.5000 


.4 


43.0084 


23.2478 


.4 


102.0703 


35.8143 


,5 


44.1786 


23.5619 


.5 


103.6689 


36.1383 


.6 


45.3646 


23.8761 


.6 


105.6833 


36.4425 


.7 


46.5663 


24.1903 


.7 


107.5132 


36.7566 


.8 


47.7836 


24.5044 


.8 


109 3588 


37.0708 


.9 


49.0167 


24.8186 


.9 


111.2202 


37.3850 


8.0 


50.2655 


25.1327 


12.0 


113.0973 


37.6991 


.1 


51.5300 


25.4469 


.1 


114.9901 


38.0133 


.2 


52.8102 


25.7611 


.2 


116.8987 


38.3274 


.3 


54.1061 


26.0753 


.3 


118.8229 


38.6416 


.4 


55.4177 


26.3894 


.4 


120.7688 


38.9557 


.5 


56.7450 


26.7035 


.5 


122.7185 


39.2699 


.6 


58.0880 


27.0177 


.6 


124.6898 


39.5841 


.7 


59.4468 


27.3319 


.7 


126.6769 


39.8982 


.8 


60.8213 


27.6460 


.8 


128.6796 


40.2124 


.9 


62.8114 


27.9602 


.9 


130.6981 


40.5265 


9.0 


63.6173 


28.2743 


13.0 


132.7323 


40.8407 


.1 


65.0388 


. 28.5885 


.1 


134.7822 


41.1549 


.3 


66.4761 


28.9027 


.2 


136.8478 


41.4690 


.3 


67.9291 


29.2168 


.3 


138.9291 


41.7832 


.4 


69.3978 


29.5310 


.4 


141.0261 


42.0973 


.5 


70.8822 


29.8451 


.5 


143.1383 


42.4115 


.6 


72.3823 


30.1593 


.6 


145.2672 


42.7257 


.7 


73.8981 


30,4734 


.7 


147.4114 


43.0398 


.8 


75.4296 


30.7876 


.8 
.9 


149.. 5712 


43.3540 


.9 


76.9769 


31.1018 


151.7468 


43.6681 



Dlam. 


Area. 


Clrcuiu. 


Dlam. 


Area. 


Clicuin. 


14.0 
.1 
.2 
.3 

.4 


153.9380 
156.1450 
158.3677 
160.6061 
162.8603 


43.9823 
44.2965 
44.6106 
44.9248 
45.3389 


8.0 
.1 

.8 
.3 
.4 


254.4690 
257.3043 
260.1553 
263.0820 
265.9044 


56..5486 
56.8628 
57.1770 
57.4911 
57.8053 


.5 
.6 
.7 
.8 
.9 


165.1300 
167.4155 
169.7167 
173.03.% 
174.3663 


4.5.5531 

45.8673 
46.1814 
46.4956 
46.8097 


.5 
.6 

.7 
.8 
.9 


268.8025 
271.7164 
274.6459 
277.5911 
280.5521 


58.1195 
58.4336 
58.7478 
59.0619 
59.3761 


15.0 
.1 
.2 
.3 

.4 


176.7)46 
179.0786 
181.4584 
183.8539 
186.2650 


47.1239 
47.4380 
47.7522 
48.0664 
48.3805 


19.0 
.1 
.2 
.3 
.4 


283.5887 
286.5211 
289.5292 
292.5530 
295.5925 


59.6903 
60.0044 
60.3186 
60.6327 
60.9469 


.5 
.6 

.7 
.8 
.9 


188.6919 
191.1345 
193.5928 
196.0668 
198.5565 


48.6947 
49.0088 
49.3230 
49.6372 
49.9513 


.5 
.6 

.7 
.8 
.9 


298.6477 
301.7i86 
304.8058 
307.9075 
311.0255 


61.8611 
61.5752 
61.8894 
68.2035 
62.5177 


16.0 
.1 
.2 
.3 
.4 


201:0619 
203.5831 
206.1199 
208.6724 
211.8407 


50.2655 
50.5796 
50.8938 
51.8080 
51.6221 


20.0 
.1 
.2 
.3 
.4 


314.1593 
317.3087 
320.4739 
323.6547 
326.8513 


62.8319 
63.1460 
63.4602 
63.7743 
64.0885 


.5 
.6 

.7 
.8 
.9 


213.8246 
216.4243 
219.0397 
221.6708 
234.3176 


51.8363 
52.1504 
52.4646 

52.7788 
53.0929 


.5 
.6 

.7 
.8 
.9 


330.0636 
333.2916 
3a6.5353 
339.7947 
343.0698 


64.4026 
64.7168 
65.0310 
05.8451 
65.6593 


17.0 
.1 
.8 
.3 
.4 


236.9801 
829.6583 
238 3522 
235.0618 
837.7871 


53.4071 
53.7212 
54.0354 
54.3496 
54.6637 


21.0 
.1 
.8 
.3 
.4 


346.3606 
349.6671 
353.9894 
356.3273 
359.6809 


65.9734 
66.2876 
66.6018 
66.9159 
67.2301 


.5 
.6 

.7 
.8 
.9 


240.5282 
243.2849 
246.0574 
248.8456 
251.6494 


54.9779 
55.2920 
55.6063 
55.9203 
56.2345 


.5 
.6 

.7 
.8 
.9 


363.0503 
366.4354 
369.8361 
373.2526 
376.6848 


67.5442 
67.8584 
68.1728 
68.48W 
68.8009 



ROGERS' DRAWING AND DESIGN. 



465 



TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. 





f ^-, — 


— 








DIanu' 


Area. 


Clrcum. 


Dlaiii. 


Area. 


Clrciini. 


23.0 


.380.1327 


69.1150 


26.0 


530.9293 


81.6814 


.1 


.<)83.5963 


69.4292 


.1 


535.0211 


81.9956 


.2 


SST.OTTO 


69.7434 


^2 


539.1287 


82.3097 


.3 


390.5707 


70.0575 


.3 


543.2521 


82.6239 


.4 


394.0814 


70.3717 


.4 


547.3911 


82.9380 


.5 


397.6073 


70.6858 


.5 


551.5459 


83.2523 


.6 


401.1500 


71.0000 


.6 


555.7163 


83.5664 


.7 


404.7078 


71.3142 


.7 


559.9025 


83.8805 


.8 


408.2814 


71.6283 


.8 


564 1014 


84.1947 


.9 


411.870t 


71.9425 


.9 


568.3220 


84.5088 


23.0 


415.4756 


72.2566 


27.0 


572.5553 


84.8230 


.1 


419.0993 


72.5708 


.1 


576.8043 


85.1373 


.2 


422.7327 


72.8849 


2 


581.0890 


85.4513 


.3 


426.3848 


73.1991 


.3 


585.3494 


85.7655 


.4 


430.0526 


73.5133 


.4 


589.6455 


86.0796 


.5 


433.7361 


73.8274 


.5 


593.9574 


86.3938 


.6 


437.4354 


74.1416 


.6 


598.2849 


86.7080 


.7 


441.1503 


74.4557 


.7 


602.6282 


87.0231 


.8 


444.8809 


74.7699 


.8 


606.9871 


87.3363 


.9 


448.6273 


75.0811 


.9 


611.3618 


87.6504 


24.0 


452.3893 


75.3982 


28.0 


615.7.522 


87.9646 


.1 


456.1671 


75.7124 


.1 


620.1582 


88.2788 


.2 


459.9606 


76.0265 


.2 


634.5800 


88.5929 


.3 


463.7698 


76.3407 


.3 


629.0175 


88.9071 


A 


467.5947 


76.6549 


.4 


633.4707 


89.2313 


.5 


471.4352 


76.9690 


.5 


637.9397 


89.5354 


.6 


475.2916 


77.2832 


.6 


642.4243 


89.8495 


.7 


479.1636 


77 ..5973 


.7 


646.9246 


90.1637 


.8 


483.0513 


77.9115 


.8 


651.4407 


90.4779 


.9 


486.9547 


78.2257 


.9 


055.9734 


90.7920 


25.0 


490.8739 


78.5398 


29.0 


660.5199 


91.1063 


.1 


494.8087 


78.8540 


.1 


665.0830 


91.4203 


.2 


498.7592 


79.1681 


.2 


669.6619 


91.7343 


.3 


502.7255 


79.48i3 


.3 


674.3565 


9?.04>'7 


.4 


506.7075 


79.7965 


A- 


678.8668 


93.3628 


.5 


510.7052 


80.1106 


.5 


683.4928 


92.6770 


.6 


514.7185 


80.4248 


.6 


688.1345 


92.9911 


.7 


518.7476 


80.7389 


.7 


692.7919 


93.3053 


.8 


522.7924 


81.0531 


.8 


697.4650 


93.6195 


.9 


526.8529 


81.3673 


.9 


702.1538 


93.9336 



nam. 


Area. 


Clrcniii. 


DIam. 


Area. 


Ctrcuin, 


30.0 
.1 
.2 
.3 
.4 


706.8583 
711.5786 
716.3145 
721.0663 
725.8336 


94.2478 
94.5619 
94.8761 
95.1903 
95.5041 


34.0 
.1 
^2 
^3 
.4 


907.9203 
913.2688 
918.6331 
934.0131 
939.4088 


106.8142 
107.1283 
107.4425 
107.7566 
108.0708 


.5 
.6 

.7 
.8 
.0 


730.6167 
735.4154 
740.2299 
745.0601 
749.9060 


95.8186 
96.1327 
96.4469 
96.7611 
97.0752 


.5 
.6 

.7 
.8 
.9 


934.8203 
940 2473 
945.6901 
951.1486 
956.6328 


108.3849 
108.6991 
109.0133 
109.3274 
109.6416 


31.0 
.1 
.2 
.3 

.4 


754.7676 
759.6450 
764.5380 
769.4467 
774.3712 


97.3894 
97.7035 
98.0177 
98.3319 
98.6460 


85.0 
.1 
.2 
.3 

.4 


963.1138 
967.6184 
973.1397 
978.6768 
984.2290 


109.9557 
110.2699 
110..5841 
110.8982 
111.2124 


.5 
.6 
.7 
.8 
.9 


779.3113 

784.2672 
789.2388 
794.2260 
799.2290 


98.9602 
99.2743 
99 5885 
99.9026 
100.2108 


.5 
.6 

.7 
.8 
.9 


989.7980 

995.3822 

10009821 

1006.5977 

10133390 


111.5265 
111.8407 
112.1549 
113.4890 
113.7832 


32.0 
.1 
.2 
.3 

A 


804.3477 
809.2831 
814 3322 
819.39f0 
824.4796 


100.5310 
100.8451 
101.1593 
101.4734 
101.7876 


36.0 
.1 
.2 
.3 
.4 


1017.8760 
1023.5387 
1029.2172 
1034.9113 
1040.6212 


113.0973 
113.4115 
113.7357 
114.0398 
114.3510 


.5 
.6 

.7 
.8 
.9 


829.5768 
834.6898 
839.8185 
844.9628 
850.1329 


102.1018 
102.4159 
102.7301 
103.0442 
103.3584 


.5 
.6 
.7 
.8 
.9 


1046.3467 
1052.0880 
1057.8449 
1063.6176 
1069.4060 


114.6681 
114.9823 
115.3965 
115.6106 
115.9348 


33.0 
.1 
.2 
.3 

.4 


855.3986 
860.4903 
865.6973 
870.9202 
876.15S8 


103.6726 
103.9867 
104.3009 
104.6150 
104.9292 


37.0 
.1 
.2 
.3 
.4 


1075.2101 
1081.0299 
1086.8654 
1092.7166 
1098.5835 


110.2389 
116.5531 
116.8672 
117.1814 
117.4956 


.5 
.6 

.7 
.8 
.9 


881.4131 
886.6831 
891.9688 
897.3703 
902.5874 


105.2434 
105.5575 
105.8717 
106.1858 
106..5000 


.5 
.6 

.7 
.8 
.9 


1104.4662 
1110.3645 
1116.3786 
1122.2083 
1128.1538 


117.8097 
118.1239 
118.4380 
118.7523 
119.0664 



466 






ROGERS' 


DRAWING AND DESIGN. 










TABLES OF AREAS AND CIRCUMFERENCES OF 


CIRCLES 


— Continued. 




Diam. 


Area.^ 


Cli'cnm. 


Dlain. 


Area. ^ 


Clrcuiu. 


Dfutn. 


Area. 


CIrcum. 


DIam. 


Area. 


CIrcum. 


88.0 
.1 
.8 
.3 
.4 


1134.1149 
1140.0918 
1146.0844 
1152.0927 
1158.1167 


119.3805 
119.6947 
130.0088 
120.3330 
120.6372 


42.0 
.1 
.3 
.3 
.4 


1385.4434 
1392.0476 
1398.6685 
1405.3051 
1411.9574 


131.9469 
133.2611 
132.5753 
132.8894 
133.2035 


46.0 
.1 

•? 
.3 
.4 


1661.9025 
1669.1360 
1676.3853 
1683.6502 
1690.9308 


144.5133 
144.8274 
145.1416 
145.4557 
145.7699 


50.0 
.1 

'.3 
A 


1963.4954 
1971.3572 
1979.2348 
1987.1280 
1995.0370 


157.0796 
157.3938 
157.7080 
158 0221 
158.3363 


.5 
.6 

•7 
.8 
.9 


1164.1564 
1170.8118 
1170.2830 
1182.3698 
1188.4724 


120.9513 
121.2655 
121.5796 
131.8938 
123.3080 


.5 
.6 
.7 
.8 
.9 


1418.6254 
1435.3093 
1432.0086 
1438.7238 
1445.4546 


133.5177 
133.8318 
134.1460 
134.4602 
134.7743 


.5 
.6 
.7 
.8 
.9 


1698.2272 
1705.5392 
1712.8670 
1720.2105 
1727.5697 


146.0841 
146.3982 
146.7124 
147.0265 
147.3407 


.5 
.6 

.7 
.8 
.9 


2002.9617 
2010.9020 
2018.8581 
2026.8299 
2034.8174 


158.6504 
158.9646 
159.2787 
159.5989 
159.p071 i 


39.0 
.1 
.2 
.3 

A 


1194.5906 
1300.7246 
1206.8743 
1813.0396 
1219.2207 


122.5221 
123.8363 
123.1504 
123.4646 
123.7788 


43.0 
.1 
.8 
.3 
.4 


1452.2012 
1458.9635 
1465.7415 
1472.5352 
1479.3446 


135.0885 
135.4026 
135.7168 
136.0310 
136.3451 


47.0 
.1 
.2 
.3 
.4 


1734.9445 
1742.3351 
1749.7414 
1757.1635 
1764.6012 


147.6550 
147.9690 
148.2833 
148 5973 
148.9115 


51.0 
.1 
.2 
.3 
.4 


2042.8206 
2050.8395 
2058.8742 
2066.9345 
2074.9905 


160.8813 
160.5354 
160.8495 
161 1637 
161.4779 


.5 
.6 

.7 
.8 
.9 


1225.4175 
1231.6300 
1237.8583 
1244.1021 
1250.3617 


124.0929 
134.4071 
1347313 
135.0354 
135.3495 


.5 
.6 

.7 
.8 
.9 


1486.1697 
1493.0105 
1499.8670 
1506.7393 
1513.6273 


136.6593 
136.9734 
137.2876 
137.6018 
137.9159 


.5 
.6 
.7 
.8 
.9 


1772.0546 
1779. .5237 
1787.0086 
1794.5091 
1803.0254 


149.3257 
149.5398 
149.8540 
150.1681 
150.4823 


.5 
.6 

.7 
.8 
.9 


2083.0723 
2u91.1697 
2099.2829 
2107.4118 
2115.5563 


161.7920 
162.1062 
162.4203 
162.7345 
163.0487 


40.0 
.1 
.2 
.3 

.4 


1256.6371 
1262.9281 
1269.2348 
1275.5573 
1281.8955 


125.6637 
125.9779 
126.3920 
126.6063 
136.9303 


44.0 
.1 
2 

'.Z 

.4 


1520.5308 
1527.4503 
1534.3853 
1541.3360 
1548.3035 


138.2301 
138.5443 
138.8584 
139.1726 
139.4867 


48.'0 
.1 
.2 
.3 
.4 


1809.5574 
1817.1050 
1824.6684 
18.S2.2475 
1839.8423 


150.7984 
151.1106 
151.4248 
151.7389 
152.0531 


52.0 
.1 
2 

!3 

.4 


2123.7166 
2131.8926 
2140.0843 
8148.2917 
3156.5149 


163.3628 
163.6770 
163.9911 
164.3053 
164.6195 


.5 
.6 
.7 
.8 
.9 


1288.2493 
1294.3189 
1301.0043 
1307.4053 
1313.8219 


127.2345 

127.5487 
137.8628 
138.1770 
138.4911 


.5 
.6 
.7 
.8 
.9 


1555.2847 
1563.2826 
1569.2963 
1576.3255 
1583.3706 


139.8009 
140.1153 
140.4293 
140.7434 
141.0575 


.5 
.6 

.7 
.8 
.9 


1847.4528 
1855.0790 
1862.7210 
1870.3786 
1878.0519 


152.3672 
152.6814 
152.9956 
153.3097 
153.6339 


.5 
.6 

.7 
.8 
.9 


8164.7537 
2173.0082 
2181.2785 
2189.5644 
2197.8661 


164.9336 
165 2479 
165.5619 
165.8761 
166.1903 


41.0 
.1 

.2 
.3 
.4 


1320.2543 
1326.7024 
1333.1663 
1339.6458 
1346.1410 


138.8053 
139.1195 
139.4336 
139.7478 
130.0619 


45.0 
.1 
3 

!3 

.4 


1590.4313 
1597.5077 
1604.5999 
1611.7077 
1618.8313 


141.3717 
141.6858 
142.0000 
142.3142 
143.6283 


49.0 
.1 
.2 
.3 
.4 


1885.7409 
1898.4457 
1901.1662 
1908.9024 
1916.6543 


153.9380 
154.2522 
154.5664 
154.8805 
155.1947 


53.0 
.1 
.2 
.3 

.4 


2206.1834 
2214.5165 
8232.8653 
2231.8298 
8239.6100 


166.5044 
166.8186 
167.1327 
167.4469 
167.7810 


.5 
.6 
.7 
.8 
.9 


1352.6520 
1359.1786 
1365.7210 
1372.2791 
1378.8529 


130.3761 

130.6903 
131.0044 
131.3186 
131.6327 


.5 
.6 

.7 
.8 
.9 


1625.9705 
1633.1255 
1640.^962 
1647.4826 
1654.6847 


142.9425 
143.2566 
143.5708 
143.8849 
144.1991 


.5 
.6 

.7 
.8 
.9 


1924.4218 
1938.2051 
1940.0042 
1947.8189 
1955.6493 


155.5088 
155.8330 
156.1378 
156.4513 
156.7655 


.5 
.6 
.7 
.8 
.9 


2248.0059 
2256.4175 
2264.8448 
2273.2879 
2281.7466 


168.0752 
168.3894 
1(58.7035 
169.017V 
169.3318 













~^''^~~~ 



















ROGERS' 


DRAWING AND DESIGN. 






467 






TART.RS OF ARFAS AND CTR.aiMFF.RF.NCF.S OF 


CIRCTP.S- Continued. 






DIaiQ. 


Area. 


Circum 


DIam. 


Area. 


CIrcuin. 


DIain. 


Area. 


circum. 


Dlam. 


Area. 


circum. 




54.0 

.1 

. .2 

.3 

.4 


8290.2210 
2398.7112 
2307.2171 
2iJ15.7386 
2324.2759 


169.6460 
169.9603 
170.2743 
170.5885 
170.9026 


58.0 
.1 
.2 
.3 
.4 


2642.0794 
2651.19T9 
2660.3331 
2669.4820 
2878.6476 


182.2134 
182.5265 
182.8407 
183.1549 
183.4690 


62.0 
.1 
.2 
.3 

.4 


3019.0705 

3028.8173 
3038.5798 
3048.3580 
3058.1520 


194.7787 
195.0929 
195.4071 
195.7212 
196.0354 


66.0 
.1 
.2 
.3 

.4 


3421.1944 
3431.5695 
3441.9603 
3452.3669 
3463.7891 


207.3451 
207.6593 
207.9734 

208.2876 
208.6017 




.5 
.6 
.7 
.8 
.9 


2332.8289 
2341.3976 
2349.9820 
2258.5821 
2367.1979 


171.2168 
171.5cl0 
171.8451 
J72.1593 
172.4735 


.5 
.6 

.7 
.8 
.9 


2687.8289 
2697.0259 
2706.3386 
2715.4670 
2724.7112 


183.7832 
184.0973 
184.4115 
184.7256 
185.0398 


.5 
.6 

,7 
.8 
.9 


3067.9616 

3077.7869 
3087.6279 
3097.4847 
3107.3571 


196.3495 
196.6637 
196.9779 
197.2930 
197.6062 


.5 
.6 

.7 
.8 
.9 


3473.3270 
3483.6807 
3494.1500 
3504.6351 
3515.1359 


208.9159 
209.2301 
209..5443 
209.8584 
210.1725 




55.0 
.1 
.2 
.3 
.4 


2375.8294 
2384.4767 
2393.1396 
2401.8183 
2410.5126 


172.7876 
173.1017 
173.4159 
173.7301 
174.0443 


59.0 
.1 
.2 
.3 

.4 


2733.9710 
2743.2466 
2753.5H78 
2761.8448 
2771.1675 


185.3540 
185.6681 
185.9833 
186.2964 
186.6106 


63.0 
.1 
.2 
.3 

.4 


3117.2453 
3127.1493 
3137.0688 
3147.0040 
3156.9550 


197.9203 
198.2345 
198.5487 
198.8628 
199.1770 


67.0 
.1 
.3 
!3 

.4 


3535.6.524 
3536.1845 
3546.7324 
3557.2960 
3567.8754 


210.4867 
210.8009 
211.1150 
211.4292 
211.7433 




.5 
.6 

.7 
.8 
.9 


2419.2227 
2427.9485 
2436.6899 
2445.4471 
2454.2200 


174.3584 
174.6726 
174.9867 
175.3009 
175.6150 


.5 
.6 

.7 
.8 
.9 


2780.5058 
2789.8599 
2799.2297 
2808.6152 
2818.0165 


186.9248 
187.3389 
18,7.5531 
187.8672 
188.1814 


.5 
.6 

.7 
.8 
.9 


3166.9217 
3176.9043 
3186.9023 
3196.9161 
3206.9456 


199.4911 
199.8053 
200.1195 
200.4336 
200.7478 


.5 
.6 

.7 
.8 
.9 


3578.4704 
3589.0811 
3599.7075 
3610.3497 
3621.0075 


212.0.575 
213.3717 
212 6858 
213.0000 
213.3141 




56.0 
.1 
.2 
.3 
A 


2463.0086 
2471.8130 
2480.6330 
2489.4687 
2498.3201 


175.9292 
176.2433 
1T6.5575 
176.8717 

177.18.i8 


60.0 
.1 
.2 
.3 

.4 


2827.4334 
2836 8660 
2846.3144 
2855.7784 
2865.2583 


188.4956 
188.8097 
189.1239 
189.4380 
189.7522 


64.0 
.1 
.8 
.3 

.4 


3216.9909 
3227.0518 
3237.1285 
3247.2222 
3257.3289 


201.0620 
201.3761 
201.6903 
202.00'44 
202.3186 


68.0 
.1 
.2 
.3 

.4 


3631.6811 
3642.3704 
3653.0754 
3663.7960 
3674.5324 


213.628S 
213.9425 
214.2.J66 
314.5708 
214.88-19 




.5 
.6 
.7 
.8 
.9 


2507.1873 
2516.0701 
2524.9687 
2533.8830 
2542.8129 


177.5000 
177.8141 
178.1283 
178.4425 
178.7566 


.5 
.6 
.7 
.8 
.9 


2874.7536 
2884.2648 
2893.7917 
2903.3343 
2912.8936 


190.0664 
190.3805 
190.6947 
191.0088 
191.3230 


.5 
.6 

.7 
.8 
.9 


3267.4527 
3277.5922 
3287.7474 
3297.9183 
3308.1049 


202.6327 
202.9469 
203.2610 
203.5752 
203.8894 


.5 
.6 

.7 
.8 
.9 


3685.2845 
3696.0.-)23 
3706.8359 
3717.6351 
3728.4500 


215.1991 

215.5133 
215.8274 
216.1416 
216.4556 




57.0 
.1 
.2 
.3 
.4 


2551.7586 
2560.7200 
2569.6971 
2578.6899 
2587.6985 


179.0708 
179.3849 
179 6991 
180.0133 
180.3274 


61.0 
.1 
.2 
.3 
.4 


2923.4666 
2933.0563 
2941.6617 
2951.2828 
2960.9197 


191.6373 
191.9513 
192.3655 
193.5796 
193.8938 


65.0 
.1 
.2 
.3 
.4 


3318.3073 
3328.5353 
3338.7590 
3349.0085 
3359.3736 


204.2035 
204.5176 
204.8318 
205.1460 
205.4602 


69.0 
.1 
.3 
,3 

.4 


3739.2807 
3750.1270 
3760.9891 
3771.8668 
3782.7603 


216.7699 

217.0841 
217.3982 
217.7124 
218.0365 




.5 
.6 
.7 
.8 
.9 


2596.7227 
2605.7626 
2614.8183 
2623.8896 
2632.9767 


180.6416 
180.9557 
181.2699 
181.5841 
181.8983 


.5 
.6 
.7 
.8 
.9 


2970.5728 
2980.2405 
2989.9244 
2999.6241 
3009.3395 


193.3079 
193.5231 
193.8363 
191.1.504 
194,4646 


.5 
.6 

.7 
.8 
.9 


3369.5545 
3379.8510 
3390.1633 
3400.4913 
3410.8350 


205.7743 
206.0885 
206.4026 
306.7168 
307.0310 


.5 
.6 

.7 
.8 
.9 


3793.6695 
3804.5944 
3815.5350 
3826.4913 
3837.4633 


218.3407 
218.6548 
218.9690 
219.2832 
219.5973 





468 



ROGERS' DRAWING AND DESIGN. 



TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. 



Dfain. 


Area. 


CIrcuin. 


Diflm. 


Arcji. 


Circuin. 


70.0 


3848,4510 


319.9115 


74.0 


4800.8408 


232.4779 


.1 


3tf59.4544 


220.3256 


.1 


4312.4721 


283.7920 


.3 


3870.4786 


230.5398 


.3 


43a4.1195 


233.1063 


.3 


3881.5084 


220.8540 


.3 


4335.7827 


233.4203 


A 


3893.5590 


331.1681 


.4 


4847.4616 


238.7345 


.5 


3903.6353 


331.4833 


.5 


4359.1563 


284.0487 


' .6 


8914.7073 


321.7964 


.6 


4870.8664 


234.3638 


.7 


8935,8049 


332.1106 


.7 


43S3.5924 


234.6770 


.8 


3936.9183 


322.4248 


.8 


4394.3341 


284.9911 


.9 


3948.0473 


232.7389 


.9 


4406.0916 


285.3053 


71.0 


3959.1921 


333.0531 


75.0 


4417.8647 


235.6194 


.1 


3970.3526 


223.3672 


.1 


4439.6585 


235.9336 


.8 


3981.5239 


233.6814 


.2 


4441.4580 


236.3478 


.3 


3992.7308 


233.9956 


.3 


4453.2783 


236.5619 


A 


4003.9384 


334.8097 


.4 


4465.1142 


236.8761 


.5 


4015.1518 


224.6239 


.5 


4476 9659 


337.1903 


.6 


4036.3908 


334.9880 


.6 


4488.8332 


387.5044 


.7 


4037.6456 


255.3522 


.7 


4500.7163 


337.8186 


,8 


4048.9160 


225..5664 


.8 


4513.6151 


238.1327 


.9 


4060.3023 


225.8805 


.9 


4524.5296 


288.4469 


73.0 


4071.5041 


226.1947 


76.0 


4536.459S 


238.7610 


.1 


4083.8317 


326.5088 


.1 


4548.4057 


339.0752 


.3 


4094.1550 


336.8330 


.3 


4560.3673 


339.3894 


.3 


4105.5040 


327.1371 


.8 


4572.3446 


3:i9.7035 


.4 


4116.8687 


327.4513 


.4 


4584.3i77 


340.0177 


.5 


4138.3491 


327.7655 


.5 


4596.3464 


340.3318 


.6 


4139.6452 


238.0796 


.6 


4608.8708 


240.6460 


.7 


4151.0571 


228.3938 


.7 


4620.4110 


240.9602 


.8 


4163.4846 


238.7079 


.8 


4632.4669 


241.2743 


.9 


4173.9379 


229.0221 


.9 


4344.5384 


3415885 


73.0 


4185.3868 


229.3363 


77.0 


4656.6357 


241.9036 


.1 


4196.8615 


329.6504 


.1 


4668.7287 


242.2168 


.3 


4208.3519 


339.9646 


.2 


4680.8474 


242.5310 


.8 


4319.8579 


330.2787 


.3 


4692.9818 


342.H451 


.4 


4231.3797 


330.5939 


.4 


4705.1319 


343.1592 


.5 


4243.9173 


230.9071 


.5 


4717.2977 


343.4734 


.6 


4354.4704 


231.3212 


.6 


4729.4793 


343.7876 


.7 


4366.0394 


381.5854 


.7 


4741.6765 


344.1017 


.8 


4277.6340 


331.8395 


.8 


4753.8894 


344.4159 


.9 


4389.2343 


232.1637 


.9 


4766.1181 


344.7301 



DIani. 


Area. 


CIrcum. 


Dlain. 


Area. 


CIreum. 


78.0 
.1 
.2 
.3 
.4 


4778.8624 
4790.6335 
4803.8983 
4815.1897 
4837.4969 


245.0442 
245.8584 
245.6725 
245 9867 
246.3009 


82.0 
.1 
.3 
.3 
.4 


5281.0173 
6393.9056 
5306.8097 
5319.7295 
5382.6650 


257.6106 
257.9247 
258.2389 
358.5531 
358.867a 


.5 
.6 

.7 
.8 
.9 


4889.8189 

4852.1584 
4864.5128 
4876.8838 
4889.2685 


346.6150 
346.9293 
247.2433 

247.5575 

247.8717 


.5 
.6 
.7 
.8 
.9 


5845.6163 
5858.5832 
5871.5658 
5384.5641 
5397.5782 


359.1814 
259.4956 
359.8097 
260.123? 
260.4880 


79.0 
.1 
.2 
.3 

.4 


4901.6699 
4914.0871 
4936.5199 
4938.9685 
4951.4338 


248.1858 
248.5000 
248.8141 
349.12^3 
249.4425 


83.0 
.1 
.2 
.3 
.4 


5410.6079 
5423.6584 
5486.7146 
5449.7915 
5462.8840 


260.7522 
361.0665 
261.3805 
361.6947 

262.0088 


.5 
.6 

.7 
.8 
.9 


4963.9137 
4976.4084 
4988.9198 
5001.4469 
5018.9897 


249.7566 

350 0708 
250.3850 
250.6991 
251.0138 


.5 
.6 

.7 
.8 
.9 


5475.9928 
5489.1168 
5502.2561 
5515.4115 
5528.5826 


262.3230 
262.6371 
262.9513 
263.2655 
263.5796 


80.0 
.1 
.3 
.3 
.4 


5036.5482 
5039.1335 
5051.7134 
5064.3180 
5076.9894 


251.3274 
251.6416 
251.9557 
252.2899 
252.5840 


84.0 
.1 
.2 
.3 
.4 


5541.7694 
5554.9720 
5568.1902 
5581.4342 
5594.6739 


263.8938 
264.3079 
364.5221 
264.8863 
265.1514 


.5 
.6 

.7 
.8 
.9 


5089.5764 
5102.3292 
5114 8977 
5127.5819 
5140.2818 


252.8983 
353.3134 
358.5265 
253.8407 
354.1548 


.5 
.6 
.7 
.8 
.9 


5607.9392 
5621.2203 
5634.5171 
5647.8296 
5661.1578 


265.4646 
265.7787 
266.0939 
266.4071 
266.7212 


81.0 
.1 
.3 
.3 

.4 


5153.9973 
5165.7287 
5178.4757 
5191.3384 
5304.0168 


254.4690 
3.i4.7833 
355.0973 
255.4115 
255.7256 


85.0 
.1 
.3 
.3 
.4 


5674.5017 
5687.8614 
5701.3367 
5714.6277 
5728.0345 


267.0354 
267.3495 
267.6637 
267.9779 
268.2930 


.5 
.6 

.7 
.8 
.9 


5316.8110 
5339.6208 
5343.4463 
5255.3876 
.5368.1446 


256.0398 
256.3540 
256.6681 
256.9823 
257.2966 


.5 
.6 

.7 
.8 
.9 


5741.4569 
5754.8951 
5768.8490 
5781.8185 
5795.3038 


268.6062 
268.9203 
269.2345 
269.5486 
269.8628 



ROGERS' DRAWING AND DESIGN. 



469 



TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. 



»lain. 


Area. 


Clrcuni. 


Diam. 


Area. 


CIrcum. 


86.0 
.1 
.2 
.3 
.4 


6808.8048 
5822.8315 
5835.8539 
5849.4020 
5863.9659 


270.2770 
270.49U 
370.8053 
271.1194 
271.4336 


90.0 
.1 
^2 
'.3 

.4 


6361.7351 

637.5.8701 
6390.0309 
6404.3073 
6418.3995 


282.7433 
283.0575 
283.3717 
283.6858 
284.0000 


.5 
.6 

.7 
.8 
.9 


5876.5454 
5890.1407 
5903.7516 
5917.3783 
5931.0206 


271.7478 
272.0619 
272.3761 
272.6902 
273.0044 


.5 
.6 

.7 
.8 
.9 


6433.6073 
6446.8309 
6461.0701 
6475.3351 
6489.5958 


284.3141 
284.6283 
284.9425 
285.2566 
285.5708 


87.0 
.1 
.2 
.3 
A 


5944.6787 
5958.3525 
5973.0430 
5985.7473 
5999.4681 


273.3186 
273.6337 
273.9469 
274.2610 
274.5753 


91.0 
.1 
.3 
.3 
.4 


6503.8833 
6518.1843 
6532.5021 
6.546.8356 
6561.1848 


285.8849 
286.1991 
286.5133 
286.8274 
287.1416 


.5 
.6 

.7 
.8 
.9 


6013.2047 
6036.9570 
6040 7250 
6054.5088 
6068.3082 


274.8894 
275.2035 
275.5177 
275.8318 
276.1460 


.5 
.6 

.7 
.8 
.9 


657.5.5408 
6589.9304 
6604 3268 
6618.7388 
6633.1666 


287.4557 
287.7699 
288.0840 
288.3983 
288.7124 


88.0 
.1 
.2 
.3 

.4 


6083.1384 
6095.9543 
6109.8008 
6133.6631 
6137.5411 


276.4602 
276.7743 
277.0885 
277.4026 
277.7168 


93.0 
.1 
.3 
.3 
.4 


6647.6101 
6662.0692 
6676.5441 
66910347 
6705.5410 


289.0265 
289.3407 
289.6548 
289.9690 
290.2882 


.5 
.6 

.7 
.8 
.9 


6151.4318 
6165.3443 
6179.3693 
6193.2101 
6307.1666 


278.0309 
278.3451 
278.6563 
278.9740 
279.2876 


.5 
.6 

.7 
.8 
.9 


6720.0630 
6734.6008 
6749.1542 
6763.7233 
6778.3083 


290.5973 
390.9115 
391.3256 
391.5398 
291.8540 


89.0 
.1 
.2 
.3 
.4 


6331.1389 
6335.1268 
6249 1304 
6363.1498 
6377.1849 


279 6017 
279.9159 
280.2301 

280 5442 
280.8584 


93.0 
.1 
.2 
.3 
.4 


6792.9087 
6807.5250 
0822.1569 
6836.8046 
6851.4680 


393.1681 
393.4833 
393.7964 
393.1106 
293.4348 


.5 
.6 
.7 
.b 
.9 


6291.3356 
6305.3031 
6319.3843 
6333.4822 
6347.5958 


281.1725 
281.4867 
281.8009 
282.1150 
282.4292 


.5 
.6 
.7 
.8 
.9 


6866.1471 
6880.8419 
6895.5524 
6910.2786 
6935.0205 


293.7389 
294.0531 
294.3673 
294.6814 
294.9956 



Dlain. 


Area. 


Clrcuin. 


DIam. 


'Area. 


Clrcimi. 


94.0 
.1 

.2 
.3 
.4 


6939.7783 
6954.5515 
6969.3106 
6984.1453 
6998.9658 


295.3097 
295.6239 
295.9380 
296.2522 
296.5663 


97.0 
.1 
.2 
.3 

.4 


7389.8113 
7405.0559 
7420.3162 
7435.5922 
7450.8839 


304.7345 
305.0486 
305.3628 
305.6770 
305 9911 


.5 
.6 

.7 
.8 
.9 


7013.8019 
7028.6538 
7043.5214 
7058.4047 
7073.3033 


296.8805 
297.1947 
297.5088 
297.8230 
298.1371 


.5 
.6 
.7 

.8 
.9 


7466.1913 
7481.5144 
7496.8532 
7.521.2078 
7527.5780 


306.3053 
306.6194 
306.9336 
307.3478 
307.5619 


95.0 
.1 
.2 
.3 

.4 


7088.2184 
7103.1488 
7118.1950 
7133.0.568 
7148.0343 


298.4513 
298.7655 
299.0796 
299.3938 
299.7079 


98.0 
.1 
.2 
.3 
.4 


7542.9640 
7558.3656 
7573.7830 
75892161 
7604.6648 


307.8761 
308.1903 
308.5044 
308.8186 
309.1327 


.5 
.6 
.7 
.8 
.9 


7163.0276 
7178.0366 
7193.0612 
7208.1016 
7223.1577 


300.0221 
800.3363 
300.6504 
300.9646 
'301.2787 


.5 
.6 
.7 
.8 
.9 


7620.1293 
7635.6095 
7651.1054 
7666.6170 
7682.1444 


309.4469 
809.7610 
310.0752 
310.3894 
310.7035 


96.0 
.1 
.2 
.3 
.4 


7238.2295 
7253.3170 
7268.4202 
7283.5391 
7298.6737 


801.5929 
301.9071 
302.2313 
302.5354 
302.8405 


99.0 
.1 
.2 
.3 

.4 


7697.6893 
7713.2461 
7728.8206 
7744.4107 
7760.0166 


311.0177 
311.3318 
311.6460 
311.9602 
312.2743 


.5 
.6 
.7 
.8 
.9 


7313.8240 
7328.9901 
7344.1718 
7359.3693 
7374.5824 


803.1637 
303.4779 
303.7920 
304.1062 
304.4203 


.5 
.6 

.7 
.8 
.9 


7775.6383 
7791.2754 
7806.9284 
7823.5971 
7838.2815 


312.5885 
312.9026 
313.2168 
313.5309 
313.8451 








100.0 


7853.9816 


314.1593 



470 



ROGERS' DRAWING AND DESIGN. 



CIRCULAR MEASURE. 

60 seconds (") make i minute ('). 

60 minutes " 1 degree (°). 

360 degrees " i circum. (C). 

The circumference of every circle whatever, is 
supposed to be divided into 360 equal parts, called 
degrees. 

A degree is -3^^ of the circumference of any circle, 
small or large. 

A quadrant is a fourth of a circumference, or an 
arc of 90 degrees. 

A degree is divided into 60 parts called minutes, 
expressed by the sign ('), and each minute is divided 
into 60 seconds, expressed by (") ; so that the circum- 
ference of any circle contains 21,600 minutes, or 
1,296,000 seconds. 



LONG MEASURE- 

12 inches = i foot. 
3 feet = I yard. 
55^ yards = i rod. 



-MEASURES OF LENGTH. 

40 rods = I furlong. 
8 furlongs = i common mile. 
3 miles = I league. 



The mile (5,280 feet) of the above table is the 
legal mile of the United States and England, and is 
called the statute mile. 



ROMAN TABLE. 



I. den 


otes One. 


XVII. denotes Seventeen. 


II. 


' Two. 


XVIII. 


' Eighteen. 


III. 


' Three. 


XIX. 


' Nineteen. 


IV. 


' Four. 


XX. 


' Twenty. 


V. 


' Five. 


XXX. 


' Thirty. 


VI. 


' Six. 


XL. 


' Forty. 


VII. . ' 


' Seven. 


L. 


Fifty. 


VIII. 


Eight. 


LX. 


' Sixty. 


IX. 


' Nine. 


LXX. 


' Seventy. 


X. 


' Ten. 


LXXX. 


' Eighty. 


XI. 


' Eleven'. 


XC. 


' Nfnety- 


XII. 


' Twelve. 


C. 


' One hundred. 


XIII. 


' Thirteen. 


D. 


' Five hundred. 


XIV. 


' Fourteen. 


M. 


' One thousand 


XV. 


' Fifteen. 


X. 


' Ten thousand 


XVI. 


' Sixteen. 


M. 


' One million. 



GREEK ALPHABET. 



A a 


alpha 


I I 


iota 


P 


P 


rho 


B ^ 


. beta 


K K 


kappa 


2 


(T 


. sigma 


r y 


. gamma 


A A . 


lambda 


T 


T 


. tau 


A S 


delta 


M 11 


mu 


Y 


V 


. upsilon 


E € 


epsilon 


N V 


nu 


4> 


<!> 


. phi 


i^ ^ 


zeta 


^ $ . 


xi 


X 


X 


. cbi 


H ^ 


eta 





omicron 


^ 


'P 


psi 


e Q 


theta 


n TT . 


pi 


fl 


b) 


. omega 



Note. — The letters of the Greek alphabet are used sometimes as 
arbitrary signs, and the letter tt (pi) is used almost universally to 
represent the ratio of the circumference to the diameter of the circle. 



ROGERS' DRAWING AND DESIGN. 



471 



TABLES 

Of Squares and Cubes, and Square and Cube Roots of 
numbers from i to 200. (See opposite column.) 

RULES. 

To And side of an inscribed square- 
Multiply diameter by 0.7071. 
Or multiply circumference " 0.2251. 
Or divide " " 4.4428. 

To £nd side of an equal square — 



Multiply diameter 

Or divide 

Or multiply circumference 

Or divide 



by 0.8862. 
" 1. 1 264. 
" 0.2821. 

" 3.545- 



Square — 

A side multiplied by 1.4142 equals diameter of its 
circumscribing circle. 

A side multiplied by 4.443 equals circumference of 
its circumscribing circle. 

A side multiplied by 1.128 equals diameter of an 
equal circle. 

A side multiplied by 3.544 equals circumference 
of an equal circle. 

A side multiplied by 1.273 equals circle inches of 
an equal circle. 



SQUARES, CUBES 


AND ROOTS. 


Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


1 


1 


1 


1.0 


1.0 


2 


4 


8 


1.414213 


1.25992 


8 


9 


27 


1.732050 


1.44225 


4 


16 


64 


2.0 


1.58740 


5 


25 


125 


2.236068 


1.70997 


6 


36 


216 


2.449489 


1.81712 


7 


49 


343 


2.645751 


1.91293 


8 


64 


512 


2828427 


2.0 





81 


729 


3.0 


2.08008 


10 


100 


1000 


3.162277 


2.15443 


11 


121 


1331 


3.316624 


2.22398 


12 


144 


1728 


3.464101 


2.28942 


13 


169 


2197 


3.6055.M 


2.35133 


14 


196 


2744 


3.741657 


2.41014 


15 


225 


3375 


3.872983 


2.46621 


16 


256 


4096 


4.0 


2.51984 


17 


289 


4913 


4.123105 


2.57128 


18 


324 


5832 


4.242640 


2.62074 


19 


361 


6859 


4.358898 


2.66840 


20 


400 


8000 


4.472136 


2.71441 


21 


441 


9261 


4.582575 


2.75892 


22 


484 


10648 


4.690415 


2.80203 


23 


529 


12167 


4.795831 


2.84386 


24 


576 


13824 


4.898979 


2.88449 


25 


625 


15625 


5.0 


2.92401 


26 


676 


17576 


5.099019 


2.962i9 


27 


729 


19683 


.5.196152 


3.0 


28 


784 


21952 


5.291502 


3.03658 


29 


841 


24389 


5.385164 


3.07231 


30 


900 


27000 


5.477225 


3.10723 


81 


961 


29791 


5.567764 


3.14138 


32 


1024 


32768 


5.656854 


3.17480 


33 


1089 


35937 


5.744563 


3.20753 


34 


1156 


39304 


5.830951 


3.23961 


35 


1225 


42875 


5.916079 


3.27106 


36 


1298 


46656 


6.0 


3.30192 


37 


1369 


50653 


6.082763 


3.33333 


38 


1444 


54872 


6.164414 


3.36197 


39 


1621 


59319 


6.244998 


3,39121 


40 1 


1600 


64000 


6.324555 


3.41995 



473 



ROGERS' DRAWING AND DESIGN. 



TABLES OF SQUARES, CUBES 



Nuirflier. 


Square. 


Cube. 


Square lioot. 


Cube Root. 


41 


1681 


68921 


6.403134 


3.44821 


42 


1764 


74088 


6.480740 


3.47603 


43 


1849 


79507 


6.557438 


3.50339 


44 


1936 


85184 


6.6J3349 


3 53034 


45 


2025 


91125 


6.708303 


3.55689 


46 


2116 


97336 


6.783330 


3.58304 


47 


2209 


103823 


6.855654 


3.60882 


48 


2304 


110592 


6.938303 


3.63424 


49 


2401 


117649 


7.0 


3.65930 


50 


2500 


125000 


7.071067 


3.68403 


51 


2601 


132651 


7.141428 


3.70843 


53 


2704 


14060W 


7.311102 


3.73251 


53 


3809 


148877 


7.280109 


3.75628 


54 


2916 


157464 


7.348469 


3.77976 


55 


3025 


166375 


7.416198 


3.80295 


56 


3136 


175616 


7.483314 


3.82586 


57 


3349 


185193 


7.549834 


3.84850 


58 


3364 


195112 


7.615773 


3.87087 


59 


3481 


205379 


7.681145 


3.89299 


60 


3600 


216000 


7.745966 


3.91486 


61 


8T21 


226981 


7.810249 


3.93649 


62 


3844 


238328 


7.874007 


3.95789 


63 


3989 


250047 


7.937253 


3.97905 


64 


4096 


263144 


8.0 


4.0 


65 


4225 


274625 


8.063357 


4.02072 


66 


4356 


287496 


8.124038 


4.04124 


67 


4489 


300763 


8.185352 


4.06154 


68 


4624 


314432 


8.216211 


4.08165 


69 


4761 


328509 


8.306623 


4.10156 


70 


4900 


343000 


8.366600 


4.12128 


71 


5041 


357911 


8.436149 


4.14081 


73 


5184 


373248 


8.485281 


4.16016 


78 


5329 


389017 


8.544003 


4.17933 


74 


5476 


405224 


8.603335 


4.19833 


75 


5635 


421875 


8.660354 


4.21V16 


76 


5776 


438976 


8.717797 


4.23582 


77 


5939 


456533 


8.774964 


4.25432 


78 


6084 


474552 


8.831760 


4.27265 


79 


6241 


493039 


8.888194 


4.29084 


80 


6400 


512000 


8.944271 


4.30887 



AND 


ROOTS- 


-Continoed. 






Number. 


Square. 


Cube. 


Square Uoot. 


Cube Boot. 


81 


6561 


531441 


9.0 


4.33674 


83 


6734 


551368 


9.055385 


4.34448 


83 


6889 


571787 


9.110438 


4.36207 


84 


70.56 


592704 


9.165151 


4.37951 


85 


7235 


614125 


9.218544 


4.39683 


86 


7396 


636056 


9.273618 


4.41400 


87 


7569 


658503 


9.327379 


4.43104 


88 


7744 


681472 


9.380831 


4.44796 


89 


7921 


704969 


9.433981 


4.46474 


90 


8100 


729000 


9.486833 


4.48140 


91 


8381 


753571 


9.539392 


4.49794 


93 


8464 


778688 


9.591663 


4.51435 


93 


8649 


804357 


9.643650 


4.53065 


94 


8836 


830584 


9.695359 


4.54683 


95 


9025 


857375 


9.746794 


4.56290 


96 


9316 


884736 


9.797959 


4.57785 


97 


9409 


913673 


9.848S57 


4.59470 
4.61043 


98 


9604 


941193 


9.899494 


99 


9801 


070399 


9.949874 


4.62606 


100 


10000 


1000000 


10.0 


4.64158 


101 


10201 


1030301 


10.049875 


4.65701 


103 


10404 


1061308 


10.099504 


4.67233 


103 


10609 


1093737 


10.148891 


4.68754 


104 


10816 


1124864 


10.198039 


4.70266 


105 


11025 


1157625 


10246950 


4.71769 


106 


11236 


1191016 


10.295630 


4.73263 


107 


11449 


1225043 


10.344080 


4.74745 


108 


11664 


1259712 


10.392304 


4.76230 


109 


11881 


1295029 


10.440306 


4.77685 


110 


13100 


1331000 


10.488088 


4.79143 


111 


12321 


1367631 


10.535653 


4 80589 


112 


12.544 


1404928 


10.583005 


4.82028 


113 


12769 


1443897 


10.630145 


4.83458 


114 


13996 


1481544 


10.677078 


4.84880 


115 


13;'35 


1520875 


10.723805 


4.86294 


116 


134.56 


1560896 


10.770329 


4.87699 


117 


13689 


1601613 


10.816653 


4.89097 


118 


13924 


1643032 


10.862780 


4.94086 


119 


14161 


1685159 


10.90871£ 


4.91868 


120 


14400 


1728000 


10.954451 


4.93242 











ROGERS' DRAWING AND 


DESIGN. 




473 












TA 


RT,RS OF 

Square Hoot, 


SQUARES, 


CUBES AND ROOTS— Continued. 














Nomber. 


Square. 


Cube. 


Cul)e Root. 




Nuinlifr. 


Square. 


Culie. 


Square Root. 


cube Root. 






121 


14641 


1771561 


11.0 


4.94608 


161 


25921 


4173281 


12.688577 


5.44012 








122 


14884 


1815848 


11.045361 


4 95967 




163 


26244 


4351538 


12.727922 


5.45136 










123 


15129 


1860867 


11.090536 


4.97319 




163 


26569 


4330747 


12.767145 


5.43255 










124 


15376 


1906634 


11.135528 


4.98663 




164 


26896 


4410944 


12.806348 


5.47370 










125 


15625 


1S53125 


11.180339 


5.0 




165 


27225 


4493125 


12.845232 


5.48480 










1S6 


15S76 


2000376 


11.224972 


5.01329 




166 


27556 


4574296 


13.884098 


5.49586 










127 


16129 


2048383 


11.269427 


5.02653 




167 


27889 


4657463 


13.923848 


5.50687 










128 


16384 


2097152 


11.313708 


5.03968 




168 


28224 


4741632 


1'3.961481 


5.51784 










129 


16641 


2146689 


11.357816 


5.05377 




169 


28561 


4826809 


13.0 


5.53877 










130 


16900 


2197000 


11.401754 


5.06579 




170 


28900 


4913000 


13.038404 


5.53965 










131 


17161 


2348091 


11.445523 


5.07875 




171 


29241 


5000211 


13.076696 


n.55049 










132 


17424 


2399968 


11.489125 


5.09164 




172 


29584 


5088448 


13.114877 


6.56139 










133 


17689 


2352637 


11.532562 


5.10446 




173 


29929 


5177717 


13.153946 


5.57305 










134 


17956 


2406104 


11.575836 


5.11723 




174 


30276 


5268024 


13.190906 


5.58377 










135 


18225 


2460375 


11.618950 


5.12992 




175 


p0625 


5359375 


12.238756 


5.59344 










136 


18496 


2515456 


11.661903 


5.14256 




176 


80976 


5451776 


13.266499 


5.60407 










137 


18769 


2571358 


11.704699 


5.15513 




177 


31329 


5545233 


13.304134 


5.61467 










138 


19044 


2638072 


11.747344 


5.16764 




178 


31684 


5639752 


13.341664 


5.62533 










139 


19321 


2685619 


11.789826 


5.18010 




179 


32041 


5735339 


13.379088 


5.63574 










140 


19600 


2744000 


11.832159 


5.19349 




180 


33400 


5833000 


13.416407 


5.64621 










141 


19881 


2808221 


11.874342 


5.20482 




181 


32761 


5939741 


13.453634 


5.65665 










142 


20164 


2863288 


11.916375 


5.21710 




182 


33124 


6038568 


13.490737 


5.66705 










143 


20449 


2924207 


11.958260 


5.23932 




183 


33489 


6138487 


13.537749 


5.67741 










144 


20736 


2985984 


13.0 


5.24148 




184 


33856 


6339504 


13.564660 


5.68773 










145 


21025 


8048625 


12.041594 


5.35358 




185 


34225 


6331625 


13 601470 


5.69801 










146 


21316 


3112136 


12.083046 


5.36563 




186 


34596 


6434856 


13.638181 


5.70826 










147 


21609 


3176528 


12.123455 


5.27763 




187 


34969 


6539303 


13.674794 


5.71847 










148 


31904 


8241792 


12.165525 


5.38957 




188 


35344 


6644672 


13.711309 


5.72865 










149 


22201 


3307949 


13.266S55 


5.30145 




189 


35721 


6751269 


13.747727 


5.73879 










150 


22500 


3375000 


12.347448 


5.31329 




190 


36100 


6859000 


13.784048 


5.74889 










151 


22801 


3442951 


13.388305 


5.33507 




191 


36481 


6967871 


13 820275 


5.75896 










152 


23104 


3511808 


13.338828 


5.33680 




192 


36864 


7077888 


13.856406 


5.76899 










153 


23409 


3581577 


13.369316 


5.34848 




193 


37249 


7189057 


13.892444 


5.778&9 










154 


23716 


3653264 


12.409673 


5.36010 




194 


37636 


7301384 


13.928388 


5.78896 










155 


24025 


3733875 


12.449899 


5.37168 




195 


38035 


7414875 


13.964340 


5.79889 










156 


24336 


3796416 


12.489996 


S.S8323 




196 


38416 


7529536 


14.0 


5.80878 










157 


24649 


3869893 


12.529964 


5.39469 




197 


38809 


7645373 


14.035668 


.5.81864 










158 


24964 


3944312 


13.569805 


5.40613 




198 


39304 


7763392 


14.071247 


5.82847 










159 


35281 


4019679 


13.609520 


5.41750 




199 


39601 


7880599 


14.106736 


5.83827 










160 


25600 


4096000 


12.649110 


5.42883 




200 


4on(>n 8000000 


14.142135 


5.84803 







474 


ROGERS' DRAWING AND DESIGN. 








UNITED 


STATES STANDARD SIZES 


OF WROUGHT IRON WELDED PIPE. 














Length of 
pipe per 


Length of 
pipe per 






Lenmh of 




No. of 




Inside 


Actual 


Thick- 


Actual 


External 


iDteroal 


square 


square 


Biternal 


Actual 


pipe con. 


We/ght 


threads 


Lengtb 




outelde 


ness. 


Inside 


circum- 


clrcdm- 


foot of 


loot of 


area, 


totoraal 


talniQg 


per foot 


per loch 


perfect 


Dom. 


Diameter. 




Diameter. 


ference. 


fereocj. 


outside 
surface 


Inside 
surface. 




area; 


one 
cubic fj'-'t. 


or length. 


of scrow. 


screw 


1 


.405 


.068 


0.269 


1.272 


0.848 


9.440 


14.15 


.129 


.0572 


2500. 


.243 


27 ■> 


0,19 


i 


.54 


.088 


0.364 


1.696 


1.144 


7.075 


10.50 


.229 


.1041 


1385. 


.422 


18 


0.29 


t 


.675 


.091 


0.493 


2-121 


1.552 


5.657 


7.67 


.358 


,1916 


751.5 


,561 


18 


0,30 


4 


- .840 


.109 


0,622 


2.652 


1.957 


4.502 


6.13 


.554 


.3048 


472.4 


.845 


14 


0,39 


\ ■ 


1.050 


.113 


0.824 


3.299 


2.589 


3.G37 


4.635 


.866 


.5333 


270,0 


1.126 


14 


0.40 


1 


1.315 


.134 


1.047 


4.1,34 


3.292 


2.903 


3.679 


1.357 


.8627 


166.9 


1.670 


lU 


0.51 


H 


1.660 


.140 


1.38 


5,215 


4.335 


2.301 


2.768 


2.164 


1.496 


96.25 


2.258 


\n 


0.54 


\h 


1.90 


.145 


1,61 


5.9G9 


5.061 


2.010 


2.371 


2.835 


2.038 


70.65 


2.694 


11} 


0.55 


2' 


2.375 


.154 


2.067 


7.461 


6.494 


1.611 


1.848 


4.430 


3.355 


42.36 


3.667 


Hi 


0.58 


2 J 


2.875 


.204 


2.467 


9.032 


7,754 


1.328 


1.547 


6.491 


4.783 


30.11 


5.773 


8 


0.89 


3 


3.50 


.217 


3.066 


10.996 


9.636 


1.091 


1.245 


9.621 


7.388 


19.40 


7.547 


8 


0.95 


3i 


4.0 


.226 


3.548 


12,566 


11.146 


.955 


1.077 


12.566 


9.837 


14.56 


9.055 


8 


1.00 


4 


4.60 


.237 


4.026 


14.137 


12,648 


.849 


0.949 


15.901 


12.730 


11.31 


10.728 


8 


1.05 


41 


5.0 


.247 


4.506 


15.708 


14.153 


765 


0.848 


1,9.635 


15.939 


9.03 


12.492 


8. 


1.10 


5 


5.563 


.259 


5.045 


1-7.475 


15.849 


629 


0.757 


24.299 


19,990 


7.20 


14,564 


8 


1.16 


6 


6.625 


.280 


6.065 


20.813 


19.054 


.577 


0.630 


.34.471 


28.889 


4.98 


18.767 


8 


1.26 


7 


7.625 


.301 


7.023 


23.954 


22.063 


.505 


0.544 


45.663 


38.727 


3.72 


23.410 


8 


1..36 


8 


8.625 


.322 


7.981 


27.096 


25.076 


.444 


0.478 


58.426 


50.039 


2.88 


28.348 


8 


1.46 


9 


9.688 


.344 


9.00 


30.433 


28.277 


.394 


a425 


73.716 


63-633 


2.26 


34.677 


8 


1,57 


10 


10.750 


.366 


10,018 


33.772 


31.475 


.355 


0.381 


90.762 


78.838 


1.80 


40.641' 


8 


1.68 


Thread taper 


three-fourths inch to one foot. 












All pipe belo 


vf lyi inches is butt-welded, and proved to 300 pou 


nds per sc 


luare inch ; i^i inch and above is lap- welded and proved || 


to 500 pounds per s 


quare inch. 













INDEX. 



Abbreviations and Conventional Signs 21-22 

Acute Angle, def 29 

Addendum Circle, desc. and illus 279-280 

Advantages of Algebra 431 

Corliss Valve Gear 390 

Logarithms, desc . 433 

Algebra — Advantages of 431 

Elements of 430 

Algebraic — def. and example 431 

Alphabet, Antique, desc. and illus. ... 52 

Greek 46S 

Altitude of a Pyramid or Cone 38 

A Triangle, def 31 

Altitude of a Triangle, def ... 31 

Aluminum, when discovered 307 

Andrews, Pres't of Nebraska Univer- 
sity, quotation 205 

Angle, Acute, def 29 

Designation of by three letters . . 29 

Def 27-29 

Obtuse and Oblique, def 29 

Right, def 29 

Of Advance of Eccentric, desc . . 372 
Angle, to divide into four parts, illus. 

and rule 88 

Of Screw Thread, desc. and illus. 234 

To transfer, illus. and rule 92 

To bisect an, illus. and rule. ... 87 

Annular Gear, desc. and illus 292 

Antimony, when discovered 307 

Antique Alphabet, desc. and illus 52 

Apex of an Angle, def 29 



PAGE 

Apex of Pyramid, def 38 

Applied Mechanics, def 212 

" Apron " of Lathe, desc 322 

Arc of a Circle, def 33 

Tangent and sine of an 34 

To find center of 97 

Areas and Circumferences of Circles, 

Tables of 461-467 

Armature, desc 399 

Construction of, desc. and illus.. 400-401 

Core, desc 400 

Disc, illus 312 

" Arrow Heads," vi'here placed 1S5 

Ash Pit of Boiler, illus. and desc 336 

Assembled Drawings, def 1S3 

Atmospheric Electricity, def 397 

Atom, def 212 

Attraction, def 207 

" Axis " used in Cabinet Projection. . . I2r 

Axiom, def 431 

Axioms, def. and examples S5-S6 

Axis of the Parabola, def 36 

Dabbit, Sectioning of 79-80 

How shown by colors 1S8 

Baffle Plates, desc 350 

" Bath " for Blue Prints, Solution for. 195 

Battery of Boilers, desc , . 350 

Beam Compasses, illus 417 

Bearing, illus 1 27 

Bearing, Self-oiling, desc. and illus. . . 403-404 

Bearings — Forms of, illus, and desc. . . 259-260 



PAGE 

" Bed " — Steam Engine, desc 364 

Of a Press, desc 308 

Bed Plate, how to draw 146 

Bell Crank, desc 390 

Belts and Pulleys, desc 266-277 

Crossed, illus 267 

Fly-wheel, illus, 3C3 

Details of 365-366 

Horse Power of, rule and ex- 
ample 269-270 

Rules, Forniuhe ami Examples 

for Speed of . . 268-270 

Bench Drill Press, desc. and illus 317 

Bending Moments, examples of, def. . 221-226 

Stresses Induced by, def 220 

Bevel Gears 282 

Desc. and illus 292-299 

How to Construct, illus. and desc. 254-297 

Bismuth, when discovered 307 

Bipolar Dynamo, def 398 

Bisect, an Angle, to 87 

A vStraight Line, to, illus 87 

Black Process Copying 192-193 

Blanking Die, illus. and desc 308 

Block Letters and Numerals 54 

Blocks, Pillow, desc. and illus 263-264 

Blow=off Pipe, desc 339 

Blue Priut, colored illustration 202 

Blue Printing, desc 189-192 

Printing, Test Pieces for 1 91-192 

" Bath " for, Solution 195 

Blue Prints, Mounting of, desc 195-196 



475 



476 



ROGERS' DRAWING AND DESIGN. 



PAGE 

Blue Prints, to make Drawings from. . 196 

Blue, Prussian, for Water Colors 188 

Boiler Bracket, illus. and desc 340-346 

Cornish, illus. and desc. . . .340, 341, 347 

Cylindrical Tubular. 340-345 

Dome, Development of, illus. , . . 175-176 

Fire Box, desc 344 

Flue, illus. and desc 340-341 

Furnace, illus. and desc 336-337 

Galloway, illus 344-348 

Lancashire, illus. and desc 341 

Locomotive, illus. and desc 344-346 

Plain Cylindrical, illus 336-337 

Slope Sheet, Development of, 

illus 176-179 

Stays, illus. and desc 340-346 

Vertical, illus. desc 346-347 

Water Tube, Dimensions of a.. . . 350-355 

Boilers and Engines, desc 335 

Battery of, desc 350 

Grate Surface of, desc 336 

Heating Surface of, desc 336 

Horse Power of, desc 354-356 

Steam Space of, desc 336 

Steam, desc 336 

Water Tube, desc. and illus 347-348 

Water Line of, desc 336 

Bolt-head, Square, desc. and illus. . . . 240 

Proportions of, desc. and illus. . . 237 

Bolt-sheets, desc 230 

Bolts, Stay, illus. and desc 346 

Stud, desc. and illus 241 

Weakest Part of 241 

Number of, for Cylinder Head . . 379 

Table of Tensile Strength of . . . . 241 

Border Lines, when to be used 423 

Bore Dividers, illus 416 

Pen and Pencil, desc. and illus.. 416 



PAGE 

Brackets— Wall, desc. and illus 262 

Brass, Sectioning of, illus 79-80 

How shown by colors. . . 188 

Brasses, Compositions for 265 

For Pillow Blocks 263, 264 

Drill Speed for 314 

Brick, Sectioning of, illus 80-81 

Bridge Wall of Steam Boiler, illus. 

and desc 336 

" Bromide " Sensitized Paper 194 

Bronze Age, Implements used in 307 

Brushes, desc 399 

Brush Holder Frame, desc. and illus. . 403, 405 

Reaction, desc. and illus 405-406 

Burlingame, L. D., quotation from 

address 196-197 

Butt-joint, illus. and desc 249 

C^abinet Projection, def 113 

Desc. and illus 121-122 

Problems in 122-127 

" Cap " Drawing Paper, size of 423 

Capital Letters, use of 53 

Carmine for Water Color 188 

Castings, how shown by colors 1S8 

Cast Iron, desc 214-215 

Sectioning of 78-80 

Factors of Safety-bar 217 

Center of a Circle, def 33 

Line, def 27 

Dead, of Steam Engine, desc. . . 364 

Chain Riveting, desc. and illus 250 

Changing Gears for Screw Cutting 322 

Chapman, Jno. G. , Quotation 

Check Nut, desc. and illus 240 

Chimney, desc 336 

Chord of a Circle, def. and illus 33 

' ' Chrome Yellow " for Water Color. . 188 



PAGE 

Circle, Arc of a, def 33 

Concentric, def 34 

Circumference of a 33 

Chord of a 33 

Diameter of a 33 

Illus. of a 33 

Segment of, def 33 

Rules Relating to 4 1-465 

Circles and their Properties 33 

Eccentric, def 35 

Sector of, def 33 

To Draw through three points. . . 98 

Circular Measure '.,. 468 

Velocity, def 211 

Pitch Circle of a Gear 279-280 

Circumference of a Circle, def 33 

Classification of Machines, desc 215 

Of Electricity 396 

" Clearance" of Die. 308 

Cleveland Twist Drill— Table of Drill 

Speed 314 

Co-abbreviation of Complement, 33 

Coefficient, def 207 

Of Safety, def 210 

Cohesion, def 207 

Coloring Drawings 18S 

Combination Die, illus 313 

Combustion, Products of 346, 347 

Commercial Rating of Engines 367 

Commutator, desc 399 

Commutator, desc. and illus 401-402 

Compasses, desc. and illus 414-415 

Beam, illus 417 

Composition for Brasses, desc 264 

Compound Winding, desc 406-407 

Compressive Strain, def .■ 217 

Concave, def 33 

Concentric Circles, def 34 



INDEX. 



477 



PAGE 

Cone, def . and illus 38 

Pulleys, desc. and illus 275, 277 

Conic Sections, def 37, 160, 161 

ConicaUhead Rivet, How to Draw, 

desc. and illus 244 

Construction — Line, def 27 

Materials for, def .- 214-215 

Of Armature, desc. and illus. . . . 400-401 

Of Commutator 401-402 

Contents, Table of xix 

Table of 21 

Conventional Method of Showing 

Square Headed Screws, illus. . 241 

Signs for Drawing Threads 235-237 

Convex, def 33 

Double, def 33 

Copper, Factor of Safety for 217 

How Shown in Colors 18S 

Copyright of Work xii 

Corliss Engine, Fishkill Landing, desc. 

and illus 388-390 

Valves, desc 388 

Valve Gear, illus 383-390 

Releasing Gear, illus 386-387 

Cornish Boiler, illus. and desc. . . .340, 341, 347 

Corollary, def 85 

Co-sine, Abbreviation of 34 

Of an Arc, illus 33 

Cotangent of a Circle, illus 33 

Countersunk-rivet, How to Draw, 

desc. and illus 244 

Coupling, Flange, illus 127 

Cover Plate Joint, illus. and desc 249 

Cranic, Rule for Finding Length of 

Stroke of 370 

Bell, illus 127, desc, 390 

Pin — Dimensions of 388 

Shaft of Steam Engine 382 



Crauk, To Draw by Isometric Projec- 
tion 119 

To Draw by Cabinet Projection. . 125 

Crosshead, desc 389 

Guides of the Steam Engine. . . . 362-364 
Of Steam Engine, desc. and illus. 382 

Pin 388 

Cube, illus. and def 37 

To Draw a, by a Cabinet Projec- 
tion 122 

Cup-head Rivet, How to Draw 243, 244 

Current Electricity, def 396 

Curved Line, def 27 

Curves and Sweeps, desc. and illus . . . 420-422 

Cycloid, The, def. and illus 2S5-286 

Cycloidal Gear Teeth, def. and illus. . . 285-293 

Rack , def 292 

Cylinder, def. and illus 37 

Of a Steam Engine, desc 362 

Of Corliss Engine, illus 384-385 

To Draw by Isometric Projec- 
tion, illus H6-11S 

To Draw a, by Cabinet Projection 123-125 
With Square Flange, How to 

Draw 145 

Walls, etc , of Steam Engine, 

Thickness of 379 

Cylinders — Development of Their In- 
tersection, desc. and illus 169-172 

How to Draw by Orthographic 

Projection 156-159 

Cylindrical Boiler, plain 336-337 

Tubular Boiler, desc. and illus. . 340, 345 
Ring, How to Draw by Ortho- 
graphic Projection 144 

Damper, Chimney, desc 336 

Dash Pot, Corliss 390 



PAGE 

Data and Rules, useful 429 

"Dead Center" of Steam Engine, 

desc 364 

Decagon, def 32 

Decimal Equivalents, Table of 457 

Equivalents of Millimeters and 

Fractions 458 

" Dedendum " Circle of Gear Wheel. . 279, 280 

Dedication by Author vii 

Definitions and Terms 27-40 

Algebraic 43 1 

Definitions and General Considera- 
tions Relating to Machine De- 
sign 207-2 1 1 

" Demy " Drawing Paper, Size of ... . 423 

" Density," def 213 

Design, Machine 205-206 

Designing a Steam Boiler, desc 350 

Machines, six points in 307 

Detailed Drawings, def 183 

Development of a Boiler Dome, illus.. 175-176 

A Four-part Elbow, illus 172-175 

A Tee Pipe 165-169 

Of the Slope Sheet of a Locomo- 
tive Boiler, illus 176-179 

Right Elbow 162-164 

Surfaces, def 113 

Surfaces, illus. and desc 162-179 

Surfaces, problems in 162-179 

Diagonal, def 32 

Stays, desc 340 

Diagram, Indicator, desc. and illus. 375, 376, 378 
Of Dimensions of Horizontal En- 
gines 369 

Of Way to Read Drawings 190 

Zenner's, illus. and desc 377 

Diameter of a Circle, illus 33 

Of Journals, example 258 



478 



ROGERS' DRAWING AND DESIGN. 



Diameter of Screw, desc 233 

" Diametral Pitch," def 280 

Die, Blanking, desc. and illus 308 

Disc, Cutting, desc. and illus. . . 313 

Male and Female, desc 308 

Dies and Presses, desc. and illus 3o8-3>5 

Blanking, illus. and desc 308 

Drawing, desc. and illus. . . .310, 312-313 

Gang of, desc. and illus 308 

Punches, Groups of 308 

Dimension — Line, def 27 

Lines 1 85- '.86 

" Dimensions " How Written on 

Drawings 185 

Of Drawings 184-187 

Of Horizontal Steam Engines . . . 369 

Of Pulleys 272-273 

Dimensions of Steam Boilers 352-354 

Directrix, def 36 

Disc Armature, note 400 

Dividers, desc. and illus 415 

Division, sign of 23 

Bow, illus 416 

Dodecagon, def 32,40 

Double Threaded Screw, desc 230 

Draft, Split, desc 341 

Drafting-room as an Interpreter to 

the Shop 197 

Drawing a Cup-Head Rivet 243-244 

A Hexagonal-nut, desc. and illus. 237-239 

Helix 228-230 

Instruments 411-426 

Linear, Subject of 25-81 

Paper ^ 422-423 

Pen, illus 416-417 

Tools, Good ones Necessary 413 

To Scale, Instructions for 424-425 

With Relation to Shop Work. . . 196-198 



PAGE 

Drawings, Working, General Subject. 181-199 

Coloring of 188 

Dimensioning of 184-187 

Marking of 184 

To Make from Blue Prints 196 

Tracing of 189-190 

Drawing-board, illus 4o-4i 

Class, Eugene C. Peck's method 

of Conducting, Note 413 

Dies, desc. and illus 310, 312-313 

Geometrical 85-1 10 

Ink 417-418 

Drilling Machines 314-317 

Drill Press — Bench, desc. and illus. ... 317 

Speeds — Table of 314 

Driving Pulleys, desc 266 

Ductility, def 207-213 

Dynamic Electricity, def 397 

Dynamics, def 212 

Dynamo — Bipolar, def 398 

Electric, Machinery, desc 393 

Illus 394, 395 

Meaning of word 398 

Multipolar, def 398 

The Electric, desc 398-399 

Unipolar, def 398 

Eccentric Circles, def. and illus 35 

Desc. 370, illus 370, 373 

Eccentric, Rule for Finding Length of 

Stroke of 370 

Strap and Rod, desc. and illus. . . 370, 374 

Eccentricity, Radius of, def 35 

" Efficiency," def 207 

Of Electric Motor, desc 399 

" Effort," def 207 

" Elasticity," def 207 

Modulus of, def 209, 217, 218 



PAGE 

Elbow, Development of a Right 162-164 

Development of Four Part, illus. 172-175 

Electric Motor, The, desc 399 

Motor, EiEciency of, desc 399 ' 

Electrical Machines, desc. and illus. . . 391-407 

Electricity, Classification of 396-398 

Def 393 

Electro-motive Force, def 397, 398 

Ellipse, def 36, 160 

Drawing an, illus 38, 106, 107 

Produced by Cutting Cone 160 

E. M. F. Abbreviation, def. . . . .Xt-.,. . . . 397 

Energy, def :..,.. 207 

Engine, Belt Fly Wheel for, illus. and 

desc 363-364 

Corliss — Cylinder of, illus 384 

Corliss Valve Gear, illus 383 

Cylinder of Corliss, illus 3S4-385 

Cylinder of Steam, desc 362 

Fishkill Landing Corliss, desc . . 388 

Lathe, desc ... 321-322 

Left-hand, desc. and illus 367 

Main Shaft of a Steam 362 

Note, Relating to Position of . . . . 368 

Reciprocating Steam, desc 362 

Right-hand, desc. and illus 367 

Rotary Steam, desc 362 

Engines and Boilers, desc 335 

Commercial Rating of 367 

Engines, Multi-cylinder Steam, desc. . 366, 367 

Newcomen, desc 335 

Overrunning, illus. and desc. . . . 367, 36S 

Right and Left Hand, illus 367 

Steam, desc 362 

Table of Dimensions of Horizon- 
tal Steam Engines 369 

The Fire and Heat, desc 335 

Underruning, illus. and desc. . . . 367, 368 



INDEX 



479 



PAGE 

Engines, Vertical, desc 368, illus. 371 

Envelope of a Solid, def 37 

Epicycloid, def. and illus 2S6, 287 

Equality, sign of 25 

Equation, def 431 

Equilateral Triangle, def 30 

To Construct 92-93 

Erasing, How Best Done 189 

Evaporation, Equivalent 355, 356 

Of Steam Boilers 350, 352 

Tensile Strength, illus 219 

Example for Figuring Engine Horse 

Power 37S 

Exercises in Geometrical Drawing. ... 87 

" pace" of Gear-tooth, desc. and illus. 279, 280 

Factor, def 207 

Of Safety, def 210 

False Perspective, def 113 

Fatigue of Metals, def 208 

Feed-pipe, desc 336 

Shaft of Lathe, desc. and illus. . 322 

Field Magnet, desc 399 

Fifteen Degree Lines, illus. and desc. . 45 

Figures, Straight-sided, defs 30 

"Finished," def 184 

Fire=box Boiler, desc 344 

Tubes, illus. and desc 340 

Flange Coupling, illus 127 

Cylinder, How to Draw 145 

With Bolts, How to Draw . 127 

" Flank" of Gear-tooth, illus. and desc. 279, 2S0 

Floor Stands or Pedestals, desc 266 

Flue Boiler, illus. and de.sc 340-341 

Fly Wheels, desc. and illus 364 

Rim Speed of 366 

Foci of an Ellipse, def 36 

Focus of the Parabola, def 36 



PAGE 

Foot, def 46S 

Sign for 132 

Force, def 207 

Electro-motive, def 397-398 

Moment of, def 220 

Formula for Estimating Horse Power 
of Crank Shaft of Steam En- 
gine 256-25S 

For Figuring H. P. of Steam 

Engine 376 

Lathe Gear Changes 326 

Size of Connecting Rod 388 

Thickness of Steam Engine Piston 3S1 

Prof. Unwin's, for Pulleys 273 

To Find the Pressure of Cross 

Head on Guide 382 

Formulae for Belt Speeds 26S 

For Screw Cutting in Lathe. 326, 32S-329 

Reading of 431-432 

Forty-five Degree Line, illus. and 

desc 45 

Foundations for Steam Engine 364 

Four Part Elbow, Development of, illus. 1 72- 1 75 
Fractions, How Placed in Dimension 

Lines 1 86 

Franklin Institute Standard Table. . . . 232 

Free-hand Lettering Specimen 55 

Friction, def 208 

Frictlonal Electricity, def 397 

Function of Slide Valve, desc 368 

Furlong, def 468 

Furnace, Boiler, illus. and desc 33^-337 

(jalloway Boiler, illus. and desc 344-348 

Gamboge for Water Color 188 

Gang Die, illus. and desc 308-312 

Gas and Vapor, Difference Between. . . 213 

Gaseous Bodies, Mechanics of 212 



PAGE 

Gauge Cock, desc. and illus 338 

Glass, desc. and illus 33S-339 

Pressure and Total Heat, Table. 356 

Steam, desc. and illus 338 

Gear — Annular, desc. and illus 292 

Teeth — Cycloidal, def. and illus. 285-293 

Involute, desc. and illus 282-2S4 

Wheels, desc. and illus 278-304 

Dimensions for, desc. and rules. 300-303 

Speeds of, rule and ex 278-280 

Trains of 304 

Gears, Bevel, def 2S2 

How to Draw, illus 292, 294-299 

For Screw Cutting in Lathe. . . . 322-329 

Rules for Pitch of 280-282 

Spur, def 282 

Worm 298, 300-302 

Def : . 2S2 

How to Draw 230 

" Gelatine " Sensitized Paper 194 

Generator — Four Pole, desc 399 

The Electric, desc 398-399 

Geometrical Drawing 85-1 10 

Exercises in 87 

Tools L'sed in 86 

Magnitudes. 27 

Proportion, sign of 24 

Glass Gauge, desc. and illus 338-339 

Grate Surface of Boilers, desc 336 

Gravity, def 208 

Greek Alphabet 468 

Handhole of Steam Boiler, illus 339-34° 

Hanger, Seller's Adjustable, illus and 

desc 260-262 

Hangers, desc. and illus 260-262 

Hawkins' Treatise on the Indicator 

Recommended 375 



480 



ROGERS' DRAWING AND DESIGN. 



" Head=stock " of Lathe, desc. and 

illus 322, 324, 325 

" Heart Wheel," Drawing a, illus no 

Heat, Total, Table of Gauge Pressure. . 356 

Heating Surface of Boilers, desc 336 

Surface of Steam Boiler's Ratio 

to Grate Surface 352 

Helix, How to Draw, desc. and illus. . . 228-230 

Heptagon, def 32 

Hexagon, def 32 

How to Construct by Instruments 50 

To Construct on a Line, illus. . . . 102 

To Inscribe in a Circle loi 

Hexagonal Nut, How to Draw, desc. 

and illus 147, 237, 239 

Prism, How to Draw 153, def. 37 

Pyramid, How to Draw 155 

Hexahedron, def. and illus 40 

Horizontal Engines, Dimensions of. .. 369 

Lines 42 

Line, def 28 

Horse Power of Belts, Rule for 269-270 

Of Steam Boiler, Rating of 350 

Of Steam Engine, Rule for Find- 
ing 376 

Of Boilers, desc 354-356 

Transmitted by Shafts 256 

How to Read Drawings, Diagram 190 

Hydraulics, def 212 

Hydrodynamics, def 212 

Hydrostatics, def 212 

Hyperbola, def 36, 161, illus. 38 

Drawing an, illus 108 

Hypocycloid, def. and illus 288-289 

Hypothenuse, def 31 

Hypothesis, def 85 

Icosahedron, def. and illus 40 



PAGE 

Illustration, Colored, Blue Print 199 

Imperial Drawing Paper, size of 423 

Inch, Sign for 132 

India Ink, desc 417, illus. 418 

How to Prepare 418 

Indicator Diagram, desc. and illus. 375, 376, 378 
Hawkins' Treatise on. Recom- 
mended 375 

" Inertia," def 208 

Injector, How to Operate 360 

Parts of, desc. and illus 359, 360, 361 

Inking, Instruction for 424 

Instruments, Drawing 411-426 

List and Selection of , . 426 

Intersection of Solids, def. of term.. . . 37 

«' Involute " Gear Teeth, def 282-284 

Iron, Cast, desc 214-215 

Drill Speed for 314 

Factors of Safety for 217 

How Shown oy Water Color. ... 188 

Meteoric, Note 307 

Wrought, desc 217 

Wrought, Factors of Safety for. . 217 

Isometric Projection, desc. and illus. . 1 14-120 

Problems in, illus 115-120 

Isosceles Triangle 30 

To Construct an 93 

Johnson, Wm. , Quotation 

Joints, Riveted, illus. and desc 245-251 

Riveted, illus. of 249-251 

Journals, desc. and example 258-259 

Diameter of, example 258 

Pressure on, example 258 

Kinematics, def 208 



PAGE 

Lancashire Boiler, illus. and desc. . . . 341 

Lap=joints, desc. and illus 249 

Lap of Valve, Outside and Inside of , . . 372 

Lathe, desc. and illus 320-322 

Engine, desc 321-322 

Formulae for Screw Cutting 

Gears 326-32S, 329 

Shafting, desc. and illus 330-332 

Lathe-speed 320, 321 

Laws, Newton's, def 214 

Of Motion, three, def 214 

Lead of Valve , 372 

Screw of Lathe, desc. and illus. . 322 

League, def 472 

Left Handed Screw, desc 230 

Hand Engines, illus. and desc. . . 367 

Lemma, def 85 

Lettering, Examples 63 

Subject Treated on 53-64 

Triangle, illus 55 

Letters, Block, and Numerals 54 

Capital, Use of 53 

Reference, When to be Used on 

Drawings 187 

Line, Broken, def 27 

Center, def 27 

Cun'ed, def 27 

Def 27 

Dimension, def :.."..' 27 

Dotted, def 27 

Dot and Dash, def 27 

Full, def 27 

Horizontal, def. . 28 

Inclined, def 28 

Irregular Curved, def 27 

Oblique, def 28 

Plumb, def. . , , "8 

Regular Curved, def 27 



INDEX. 



481 



PAGE 

Line, Right, def 27 

Shade, def 27 

To Divide a Straight Line 91 

Vertical, def 28 

Waved, def 27 

Linear Drawing, Subject of 25, 81 

Velocity, def 211 

Lines, Border, When to be Used 423 

Fifteen Degree, illus. and desc. . 45 

Forty-five Degree, illus. and desc. 45 

Parallel, def 28 

Seventy-five Degree, illus. and 

desc 45 

Shade, desc. and illus 65-73 

Sixty Degree, illus. and desc .... 45 

Thirty Degree, illus. and desc. , . 45 

To Draw Parallel 90 

Vertical and Horizontal, def. ... 44 

Liquid, def 213 

Load, def 20S-209 

Locomotive Boiler, illus. and desc. . . . 344-349 

Logarittimic Table 433 

Table— Use of 433-434 

Logarithms, desc 433-434 

Rules for Application 434 

Tables of 435-456 

Advantages of 433 

Long Measure, rule 468 

iVlachine Design 205 

Man as a, Note 216 

Modulus of a, def 209 

Punching and Shearing 313-315 

Tool Pullej's, speed of 270 

Machines, Classification of, desc 215 

Desc 215-216 

Drilling 314, 316-317 

Electrical, desc. and illus 391-407 



Machines, Metal Working, desc. and 

illus 307-332 

Milling, illus, and desc ....316, 318-319 

Six Points in Designing 307 

Magnetic Field, desc 399 

Magneto Electricity, def 397 

Main Shaft of the Steam Engine, desc. 362 

Male and Female Die, desc 308 

" Man as a Machine, " Note 216 

Manganese, When Discovered 307 

Manhole, desc. and illus 339 

Masonry, Factors of Safety for 217 

" Mass," def 213 

Materials for Construction, def 214-215 

Strength of, def 210 

Matter, Properties of, def 212-213 

Three States of, def 213 

McWhinney, Quotation by Prof 

Mean Effective Pressure, Rule for 

Finding 377 

Measure, Circular 46S 

Long 468 

Mechanics, Applied, def 212 

Squares, Note 29 

Theoretical 205 

Mechanism, Theory of 205 

" Medium " Drawing Paper, Size of. . 423 
Metal Working Machines, desc. and 

illus 307-332 

Metals, Discovery of, desc 307 

Fatigue of, def 208 

Meteoric Iron, Note 307 

Mile, Common, def 468 

Milling Cutter, illus 127 

Machines, illus. and desc. . .316, 318-319 

Machine, Vertical Spindle, illus. 318-319 

Minutes, Part of a Circle, 34 

Mixed — Line, def 28 



PAGE 

Modulus, def 209 

Of Elasticitj', def 209, 217-218 

Of a Machine, def 209 

Of Resistance, def 209 

Of Rupture, def 209 

Section, def 222-223 

Molecule, def 212 

Moment, def 209 

Difference Between Weight and. . 213 

Of Force, def 220 

Momentum, def 209 

Motion, def 209-210 

Three Laws of, def 214 

Motor Electric, Efficiency of, desc. . . . 399 

The Electric, desc 399 

Mounting Blue Prints, desc 195-196 

Mud Drum, desc. and illus 348 

Multi-cylinder Engines, desc. and 

illus 366-367 

Multiplication, Sign of 23 

Multipolar Dynamo, def 398 

Illus 394-395 

Negative Electricity, def 396-398 

Quantity, def 431 

Newcomen Engine, desc 335 

Newton's Laws, def 214 

Laws, What they tell us 214 

Nickel, Note Relating to 307 

Plated Sheet Steel Scale, desc. . . 425 

When Discovered 307 

Nozzles of Steam Injector, desc 360 

Numerals, Roman 46S 

Nut, Square, desc. and illus 240 

Check, desc. and illus 240 

Hexagonal, How to Draw 147 

Nuts, Proportions of, desc. and illus. . 237 



482 



ROGERS' DRAWING AND DESIGN. 



PAGE 

Oblique Angle, def 29 

Line, def 28 

Objects, Orthographic Projection 
of 149-159 

Obtuse Angle, def 29 

Octagon, def 32 

How to Construct by Instruments 51 

To Describe on a Line, illus 103 

To Describe in a Square, illus.-. .• - IC2 
To Inscribe in a Circle, illus. . . . 105 

Octahedron, def. and illus 40 

Operation, Algebraic, example 431 

Of Slide Valve, desc. and illus . . 370 

Orthographic Projection, desc. and 

illus 113, 128-161 

Problems in, illus - 132-159 

Of Oblique Objects, illus. ..... 149-159 

Oval, to Draw by Circular Arcs, illus. . 105 

r aper, Size for Patent Drawings 423 

Parabola, Drawing a, illus 107 

Illus. 36-38 160-161 

Parallel Lines, def 28 

To Draw a, illus. and rule 90 

Parallelogram, def .• 31 

To Construct a 96-97 

Patent Office, Size of Official Drawing 

Paper 423 

Peck's, Eugene C, M. E., Method of 
Conducting a Drawing Class, 

Note 413 

Pedestals and Pillow Blocks, desc. 

and illus ; 263-266 

Pen, Drawing, illus 416-417 

Made for Round Writing, illus . . 56 

Pencil — Bow, illus 416 

Penciling, Instruction for 424 

Pencils, Hard and Soft, How to Sharpen 422 



PAGE 

Pentagon, def 32 

To Inscribe in a Circle, illus. ... 104 

Pentagonal Prism, def 37 

Perpendicular — Line, def 29 

Line, to Draw a, illus. and rule. . 89 

Perspective, False, def 113 

Physics, Object of Study of 212 

Def 212 

Pillow — Blocks and Pedestals, desc. 

and illus 263, 264 

Brassesfor 263, 264 

Pipe — Development of Tee 165-169 

How to Draw a, by Orthographic 

Projection 142 

Sizes, Table of Standard 459 

Piston — Area, Rule for Finding 377 

Rod of Steam Engine 362, 3S0-3S2 

vSteam, desc 3S0 

Pitch Circle, illus. desc 279, 280 

" Pitch " of Rivets, desc. and illus. . . . 246 

Of Screw, desc . 232 

Of Screw, Rule How to Find. . . . 233 

Pivots and Journals, Pressure on 25S, 259 

Plan of the Work xvii 

Plane Figure, def 30 

" Platen " of the Milling Machine 318 

Plumb — Line, def 28 

Pneumatics, def - 212 

Point, def 27 

Pole=pieces, desc 399 

Polygon, def 30 

To Draw on a Line, illus 103 

To Inscribe in a Circle, illus . . . 104 

Polygons, Note Relating to. 32 

Polyhedron, def. and illus 40 

" Pores," def 213 

Positive Electricity, def 396, 398 

Quantity, def 431 



PAGE 

Power, def 210 

Sources of, desc 255 

Preface xv-xviii 

Presses, Dies and, desc. and illus 308-315 

Prime Movers, Useful Work of 255 

Principle of Work -211 

Printing, Blue 189-192 

Frame, desc. and illus 190 

Frames, Note 191 

Paper, Sensitizing of 193-195 

Prism, Hexagonal, def 37 

How to Draw 153 

How to Draw by Orthographic 

Projection 132, 148-149 

Pentagonal, def 37 

Quadrangular, def 37 

Prisms, illus 37 

Triangular. 37 

Problem, def 85, 431 

Problems in Cabinet Projection, illus. 122-125 

In Development of Surfaces. . . 162-179 

In Isometric Projection, illus. . . 115-120 

In Orthographic Projection, illus 132-159 

Projection, Cabinet, desc. and illus. . . 121-127 

Isometric, desc. and illus 1 14-120 

Orthographic, desc. and illus. . . 128-161 
" Projections," General Subject of. . . 113-179 

Properties of Circles 33 

Of Matter, def 212-213 

Proportion of Bolt-heads, desc. and 

illus ; 237 

Of Nuts, desc. and illus 237 

Proportions for Arms of Gear Wheels. 300-303 

" Proposition," def 85 

Protractor, illus. and desc 420 

Prussian Blue for Water Colors 188 

Pulleys, Anns of, desc. and illus 274 

Cone, desc. and illus 275-277 



INDEX. 



483 



PAGE 

Pulleys, Crowning of, desc 272 

Dimensions of, ill us. and example 272-273 

Proportions for Arms of ... . . 274 

Proportions for Bulbs of 272-273 

Rules, examples and illus. of . , - . 271-277 

Step Cone, desc. and illus, 277 

Thickness of Rims of 272 

Ti£;lit and Loose, desc 273 

Punch and Die, illus. and desc 308 

Punching and Shearing Machine 313-315 

Pyramid, Hexagonal, How to Draw. . . 155 

Illus 38-39 

Quadrangular Prism, def 37 

Quadrant, def 34 

Quadrilateral Figure, def 31 

Quadrisect an .\ngle, to, illus. and 

rule 88 

Quotation from American Machinist . . vii 

Jno. G. Chapman 

L. D. Burlingame Relating to 

Drafting Room and Shop .... 196-197 

Opposite Title Page xi 

President Andrews Relating to 

Machine Design 205 

Prof. McAMiinney 

Will. Johnson 

Raabe's, H.E. , List of Drawing Instru- 
ments 426 

Rack — Cycloidal, desc 292 

Radiated Electricity, def 396 

Radii of a Circle, def 33 

Radius of a Circle, def. and illus 33 

Eccentricity, def 35 

" Ram " of a Press, desc 30S 

Ratio Between Heating and Grate Sur- 
faces 352 

Velocity, def 211 



PAGE 

Reading of Formulae 431-432 

Working Drawings 198 

Reciprocating Steam Engine, desc 362 

Rectangle, def 31 

To Construct a, illus. and rule . 96 

Rectilinear Figure, def 30 

Red, Vermilion, for Water Color iSS 

Reference Letters, When to be I'sed 

on Drawings 187 

Resistance, Modulus of, desc 209 

Theoretical and Practical 209 

Rhomboid, def 31 

Rhombus, def 31 

Right .\ngle, def 29 

Angled triangle, def 31 

Handed Screw, desc 230 

Hand Engines, illus. and desc . 367 

Line, def 27 

To Trisect, illus. and rule 88 

Ring, Cylindrical, How to Draw 144 

Rivet, Length of 245 

Riveted Joints, Breaking of 245 

Illus. and desc . . 245-251 

Strength of 247-249 

Riveting, Chain, desc. and illus 250 

Punching Holes for 243 

Rivets and Joints 243-251 

Diagonal Pitch of, desc 247 

Pitch of, desc. and illus 246 

Staggering of, desc. and illus. . . 250 
Robinson's, A.W., Office Rules, Quota- 
tion from 1 87 

Rod, def 468 

Roebling, Statement by Chas. G 15 

Roman Numerals, def. and illus 468 

Root, Sign of 24 

Rotary Steam Engine, desc 362 

"Rotary Table" of the Milling Ma- 
chine 318 



PAGE 

Rouillon's, Louis, List of Drawing 

Instruments 426 

Round-Headed Screws, desc. and illus. 241 

Writing, Specimen 56 

Rule for Finding the Area of a Steam ■ 

Piston 377 

Diameter of Piston Rod 381-382 

Finding the Mean Effective Pres- 
sure on Piston. . 377 

Horse Power of Steam Engine. . 376 

Horse Power of Belts 269 

How to Find " Pitch " of Screw. 233 

How to Use Logarithms 433, ex. 434 

To Find Length of Stroke Crank 

and Eccentric 370 

Rules and Data, Useful 429 

And Examples for Safety Valve. 357-358 

And Scales, desc. and illus 419 

For Application of Logarithms. . 434 
For Finding Dimensions of 

Gear Wheels. 300-303 

Horse Power of Shafts 257 

Pitch of Gears 280-2S2 

Proportioning Pulleys 268-269 

Relating to Circle 461 

Relating to Square 469 

Speed of Driver and Follower 

Pulleys 268, 269 

Ruling Pen. ilUis 416 

" Running Under " Engine, desc. and 

illus 367-36S 

Rupture, Modulus of, def 209 

" Saddle " of the Milling Machine. . . 318 

Safety, Coefficient of, def 210 

Factor of, def 210 

Valve — Rules and Examples for. 357-35S 

Scale, illus 20 

Flat Box-wood, desc 426 



484 



ROGERS' DRAWING AND DESIGN. 



PAGE 

Scale, Nickel Plated Sheet Steel, desc. 425 

Used in Lettering 53 

Scales, Rules and, desc. and illus 419 

Scholium, def 85 

Screw Cutting, Changing Gears for. . . 322 

Diameter of, desc 233 

Double Threaded , desc 230 

Gears for Lathe 322-329 

I^eft-handed, desc 230 

" Pitch " of, desc 232 

Right-handed, desc 230 

Set, desc. and illus 240-241 

Single-threaded, desc 230 

Thread — Angle of, desc. and illus. 234 

Threads — Conventional Signs for 

Drawing 235-237 

Threads, Note 230 

Threads, Table of IT. S. Standard. 232 

Triple-threaded, desc 232 

Screws and Bolts 228-241 

Round-headed, desc. and illus. . . 241 

Seconds (Part of a Circle), def 34 

Section Lining, illus. and desc 77-8' 

Modulus, def 222-223 

Sectioning Metals, etc 78-80 

Sector of a Circle, illus. and def 33 

Segment of a Circle, def 33 

Seller's Adjustable Hanger, illus. and 

desc 260-262 

Screw Thread, desc 232 

Semi°circle, def 33 

Sensitizing of Printing Paper 193-195 

" Sepia " for Water Colors 18S 

Series Winding, desc. and illus 406, 407 

Set=screw, desc. and illus 240-241 

Squares, illus 43-44 

Use of 45-48 

Shade Line, Specimens of 65-73 



Shading, Parallel Line, illus. and desc. 74-77 

" Shaft-feed " of Lathe, desc 322 

Shafting Lathe, desc. and illus 330-332 

Proper Speed of 258 

Rviles for Horse Power of 257 

Shafts and Shafting 256-258 

Formulae for Strength of 257-258 

Horse Power Transmitted by 256 

Strains Produced in, desc 256 

" Shank " of a Punch, desc 308 

Shearing Strain, def 217 

Strength, def 220 

Sheet Metal, Sectioning of, illus So, 81 

Shop Work, Drawing withtRelation to . 196-198 

Shunt Winding, desc. and illus 406-407 

" Sienna," Raw, for Water Color 188 

Signs, Conventional 21 -22 

For Designing Screw Threads. . . 235-237 

Sine of an Arc, def 34 

Single Threaded Screw, desc 230 

Sixty Degree Lines, illus. and desc ... 45 

Slide Valve, Function of, desc 36S 

Operation of, desc. and illus. . . . 370-375 

Smoke^box, desc 340 

" Snail," Drawing a, by Circular .\rcs, 

illus 109 

Solid, def 27-37 

A, def 213 

"Solution" for Bath Used in Blue 

Printing 193 

For Sensitizing Paper, Recipe.. . 192-193 

Specimens of Lettering 53-64 

Speed Lathe, illus. and desc 320-321 

Of Machine Tool Pulleys 27a 

Shafts, Proper, desc 25S 

Gear Wheels, rule and example. 278-280 

Sphere, def. and illus 40 

Spiral, Drawing a, illus. loS-iog 



PAGE 

Spur Gears, def 282 

Square, def 31 

Headed Screws — Conventional 

Method of Representing, illus. 241 
How to Construct by Instru- 
ments 48-49 

Nut, desc. and illus 240 

Rules Relating to 469 

Thread, def 230-234 

To Construct a, illus. and rule. . 95 

To Describe about a Circle, illus. 

and rule \. . . . . loi 

To Inscribe in a Circle,\illus. 

and rule 100 

Squares and Cubes and Square and 

Cube Roots, Tables of 469-471 

Mechanics', Note 29 

Staggering of Rivets, desc. and illus.. 250 

Standards, Brasses for, desc. and illus. 265 

Static Electricity, def 396 

Statics, def 212 

Stay Bolts, illus. and desc : 346 

Stays, Boiler, illus. and desc 340-346 

Through and Diagonal 340 

Steam Boilers, desc 336 

Boilers, Designing a, desc 350-354 

Boilers, Evaporation of 350-352 

Chest, desc 379 

Engine, Formula for Strength of 

Shaft 256 

Engine, Horse Power, Example 

of Figuring 378 

Engine, Parts of 362-364 

Engines, desc 362 

Gauge, desc. and illus 338 

Piston, Rule for Finding Area of. 377 

Piston, desc 380 

Ports, desc 380 



INDEX. 



485 



steam Rating of Horse Power 350 

Space of Boilers, desc 336 

Total Heat Units in, Table 356 

Works, illus ix 

Steel, desc 215 

Factors of Safety for 217 

How Shown by Water Color. . . . 188 

Sectioning of, illus 79, 80 

Soft, Drill Speed for 314 

Stone, Sectioning of, illus 80, 81 

Stop=clutch, desc. and illus 314 

Straight Line — to Bisect a 87 

To Divide a, illus. and desc gi 

Strain, def 210 

Strains Produced in Shafts, desc 256 

Tensile, def 217 

Strengtli of Materials, def 210 

Riveted Joints 247-249 

Shearing, def 220 

Tensile, def 210 

Ultimate, def 210, 218 

Stresses, def 210, 217-228 

Induced by Bending, def 220 

•• Stripper " of a Die, illus 30S 

Stud Bolt, desc. and illus 241 

Stuffing Box of the Steam Engine. . . 362 

Surface, def 27 

As a Magnitude 30 

Surfaces, Development of, illus. and 

desc 162-179 

1 able. Logarithmic, begin at page 435 

Example of Use of 433 

Of Contents. xix 

Of Contents 21 

Of Decimal Equivalents of Milli- 
meters and Fractions 458 

Of Decimal Equivalents, J" iV ■ 457 



PAGE 

Table of Standard Wire Gauges 464 

Of Standard Pipe Sizes ....... 459 

Diameter of Rivets and Thickness • 

of Plates 249 

Drill Speeds 314 

Evaporation of Coal 352 

Dimensions of Horizontal Steam 

Engines 369 

Tensile Strength of Bolts 241 

Total Heat Units in Steam 356 

U. S. Standard Threads 232 

Tables of Logarithms 435-460 

Of Squares and Cubes and Square 

and Cube Roots 469-471 

And Index 427-485 

Of Areas and Circumferences of 

Circles 461-467 

Tail-stock of Lathe 322-327 

Tangent, Abbreviation of 34 

Of an Arc, def 34 

To Draw to a Circle, illus. and 

rules 98-100 

Of a Circle, illus 33 

Tee-square, illus. and desc 42 

1 enacity, def 210 

Tensile-strength — Table for Bolts. . . 241 

Illus 219 

Def.. . . 210 

Tensile Strain, def 217 

Terms and Definitions 27-40 

Test Pieces for Blue Printing 191-192 

Tetrahedron, def. and illus 40 

Theorem, def 85, 431 

Theoretical Mechanics, def 205 

Resistance, def 210 

Theory of Mechanism, def 205 

Thread — Square, desc 234 

V desc. 230, illus. 231, 230 

Thirty Degree Lines, illus. and desc. . 45 



PAGE 

Through Stays, desc 340 

Thumb=Tacks, How Secured to Board 423 

Tight and Loose Pulleys, desc 273 

Timber, Factors of Safety for 217 

Tints and Colors 188 

Tool Chest, to Draw by Isometric Pro- 
jection 119 

Cabinet Projection 126 

TooURest of Lathe, desc. and illus. . . 323 

Tools Used in Geometrical Drawing. . . 86 
Tracing Cloth, Smooth and Dull Side of 189 

Of Drawings 189-190 

Tracings, Order to be Followed in 

Making Lines i S9 

Trains of Gear Wheels 304 

Transferring an Angle, illus. and rule 92 

Transmission, def 255 

Trapezium, def 31 

Trapezoid, def 31 

Triangle, Altitude of a 31 

Def 30 

How to Construct by Instruments 50 

Illus 43-44 

To Construct a, rule and illus. . . 94 

Triangles 30 

Triangular Prism 37 

Triple Threaded Screw, de.sc 232 

Trisect a Right Angle, to, illus. and 

rule 88 

Truncated Pyramid, illus 38-40 

Tubes, Fire, illus. and desc. ........ 340 

Ultimate Strength, def 210-218 

Unipolar Dynamo, def 398 

I'nit Stress and Strain, def 21S 

Un-vWn's, Prof., Formula for Belts. . . . 273 

Use of Logarithmic Table 433-434 

U. S. Standard Screw Threads 232 



486 



ROGERS' DRAWING AND DESIGN. 



PAGE 

Valve, Functions of, desc 368 

Valves, Corliss, desc 388 

Gear, Corliss, desc. and illus. . . . 388-390 

Corliss, desc. and illus 3S3 

Mechanism of the Steam Engine 362 

Steam Engine, illus. and desc . . . 376 

Safety, rule 357-358 

Vapor, Difference Between Gas and. . . 213 

Velocity, def. and ratio 210, 211 

Circular, def 211 

Linear, def 211 

Ratio, def 211 

Vermilion, for Water Color 188 

Vertex of an Angle, def 29 

Vertical Boiler, illus. and desc. . . .346, 347, 351 

Engines, desc 36S, illus. 371 

Lines, def 28, 44 

Spindle Milling Machine, illus. . 318, 319 
"View" in Drawing to be Drawn First. 153 

Vis=viva, def 211 

Voltaic Electricity, def 397 



PAGE 

" Volume," def 213 

V — Thread, desc. 230, illus. 231 230 

^Vater — Column, desc. and illus 339 

Leg of Boiler, desc 344 

Line of Boilers, desc 336 

Tube Boiler, illus 354-355 

Boiler, Dimensions of 350 

Boilers, desc. and illus 347- 348 

Wall-brackets, desc. and illus 262 

Waved Line, def 27 

Wedge, How to Draw 147 

How to Draw by Orthographic 

Projection 132 

Weight and Moment, Difference Be- 
tween 213 

Wheel, Belt, Fly, illus 363 

Winding Series, desc 406, 407 

Shunt, desc. and illus 406-407 

Wire Gauges, Table of Standard 460 

Wood, Sectioning of, illus 80 



PAGE 

Work, def 211 

Explanatory Note 211 

Preparing for 423-424 

Principle of 211 

Working Drawings, General Subject.. 181-199 

Reading of 198 

Worm Gears 298, 300-302 

Def 282 

Wrench — Proportioning of, desc. and 

illus 242 

Use of, for Tightening Bolts, desc. 242 

Wrist— Plate, desc 390 

Wrought Iron, desc ip 215 

Factors of Safety for. 217 

Sectioning of, illus .".... 78-80 

Yard, def 468 

Yellow, Chrome, for Water Color 188 

Aeuner's Diagram, illus. and desc. . . 377-378 



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A GOOD BOOK |S A GOOD FRIEND 

To Our Readers : 

The good books, here described, de- 
serve more than a passing notice, consider- 
ing that the brief description under each 
title indicates only their wide scope, and is 
merely suggestive of the mine of useful infor- 
mation contained in each of the volumes. 

Written so they can be easily under- 
stood, and covering the fundamental prin- 
ciples of engineering, presenting the latest 
developments and the accepted practice, giv- 
ing a working knowledge of practical things, 
with reliable and helpful information for 
ready reference. 

These books are self-educators, and "he 
who runs may read" and improve his pres- 
ent knowledge in the wide field of modern 
engineering practice. 

Sincerely, 

Theo. Audel & Co. 

Publishers 
72 5th Ave., N. Y. 

BOOKS THAT WILL ANSWER YOUR QUESTIONS 



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Hawkins' Mechanical Dictionary. 

If the reader often encounters words and 
allusions whose meaning is not clear, or is a 
busy man, without time to wade through a 
whole volume on any subject about which in- 
formation is desired, or is a student or a profes- 
sional man, feeling the frequent need of a first 
class reference work, he will readily appreciate 
the qualities that are to be found in this most 
useful book. 

It is a cyclopaedia of words, terms and phrases 
in use in the Mechanic Arts, Trades and Scien- 
ces — "many books in one." It is the one book 
of reference no student or expert can dispense 
with. 704 pageSj 6^x8^ inches, handsomely 
bound. 

Hawkins' Electrical Dictionary. 

A reliable guide for Engineers, Contractor';, 
Superintendents, Draftsmen, Telegraph and 
Telephone Engineers, Wire and Linemen. 

Contains many books in one, and is an en- 
tirely new and original work. Clearly and 
plainly defining the full use and meaning of the 
thousands upon thousands of words, terms and 
phrases used in the various branches and de- 
partments of Electrical Science. 

No Dictionary has to the knowledge of the 
publishers been printed to date that has kept 
pace with the rapid development of Electrical 
Engineering. 

It measures 6^x8^ inches, is over one and 
one-half inches thick; the book weighs about 
two and one-half pounds, giving a finish to the 
book which is charming. 



$3^ 



$3^ 



Rogers' Erecting and Operating> 

Provides full information on methods success- 
fully proved in modern practice to be the best 
for use in erecting and installing heavy machin- 
ery of all kinds. It includes a systematic 
course of study and instruction in Mill Engi- 
neering and Millwrightingy together with 
time saving tables, diagrams and a quick refer- 
ence index. 

Rogers' Drawing and Design. 

This volume is arranged as a comprehensive, 
self-instruction course for both shop and 
draughting room. 

Contains 506 pages, illustrated by over 600 
cuts and diagrams, very many of them full page 
drawings; the book is printed on a very fine 
grade of paper; it measu-res S^^xioJ^ inches 
and weighs over 3 pounds; it is in every way 
completely up-to-date. 

Rogers' Machinists Guide. 

Is a manual for Engineers and Machinists on 
modern machine shop practice, management, 
grinding, punching, cutting, shearing, bench, 
lathe and vise work, gearing, turning, and all 
the subjects necessary to advance in shop 
practice, and the working and handling of 
machine tools. The most complete book on 
these subjects. Fully illustrated. 

Audei's Answers on Practical 
Engineering. 

Gives you the everyday practice and simple 
laws relating to the care and management of a 
steam plant. Its 250 pages make jou familiar 
with steam boi'ers, steam, fuel, heat, steam 
gauge, installation and management of boilers, 
pumps, elevators, heating, refrigeration, engi- 
neers' and firemen's law, turbines, injectors, 
valves, steam traps, bells, gears, pulleys, 
electricity, etc., etc. 



$2 



$2 



$2 



$1 



Homans^ Automobiles. 

"Homans' Self Propelled Vehicles'* gives full 
details on successful care, handling, and how to 
locate trouble. 

Beginning at the first principles necessary to 
be known, and then forward to the principles 
used in every part of a Motor Car. 

It is a thorough course in the Science of 
Automobiles, highly approved by manufacturers, 
owners, operators and repairmen. Contains 
over 400 illustrations and diagrams, making 
every detail clear, written in plain language. 
Handsomely bound. 

Audels Answers on Automobiles. 

Written so you can understand all about the 
construction, care and management of motor 
cars. The work answers every question that 
may come up in automobile work. The book 
is well illustrated and convenient in size for the 
pocket. 

Audels Gas Engine Manual. 

This volume gives the latest and most 
helpful information respecting the construction, 
care, management and uses of Gas, Gasoline 
and Oil Engines, Mari-ne Motors aftd 
Automcbile Engines, including chapters on 
Producer Gas Plants and the Alcohol 
Motor. 

The book is a practical educator from cover 
to cover and is worth many times its price to 
any one using these motive powers. 

Audels Answers on Refrigeration (2VoIs.) 

Gives in detail all necessary information on 
the practical handling of machines and appli- 
ances used in Ice Making and Refrigeration. 
Contains 704 pages, 250 illustrations, and writ- 
ten in question and answer form; gives the last 
word on refrigerating machinery. Well bound 
and printed in two volumes ; complete. 



$2 



$|--55 



$2 



$4 



Homans' Telephone Engineering. 

Is a book valuable to all persons interested 
in this ever-increasing industry. No expense 
has been spared by the publishers, or pains by 
the author, in making it the most comprehen- 
sive hand-book ever brought out relating to the 
telephone, its construction, installation and suc- 
cessful maintenance. Fully illustrated with 
diagrams and drawings. 

Rogers^ Pumps and Hydraulics (2 Vols.) 

This complete and practical work treats ex- 
haustively on the construction, operation, care 
and management of all types of Pumping Ma- 
chinery. The basic principles of Hydraulics 
are minutely and thoroughly explained. It is 
illustrated with cuts, plans, diagrams and draw- 
ings of work actually constructed and in opera- 
tion, and wll rules and explanations given are 
those of the most modern practice in successful 
daily use. Issued in two volumes. 



Hawkins^-Lucas* Marine Engineering. 

This treatise is the most complete published 
for the practical engineer, covering as it does a 
course in mathematics, the management of ma- 
rine engines, boilers, pumps, and all auxiliary 
apparatus, the accepted rules for fguriruj 
tlie safety-valve. More than looo ready 
references, 807 Questions on practical ma- 
rine engineering are fully answered atid 
explained, thus forming a ready guide in solv- 
ing the difficulties and problems which so often 
arise in this profession. 



$1 



$4 



$2 



Hawkins* Electricity for Engineers. 

The introduction of electrical machinery in 
almost every power plant has created a great 
demand for competent engineers and others 
having a knowledge of electricity and capable of 
operating or supervising the running of electrical 
machinery. To such persons this pocket-book 
will be found a great benefactor, since it con- 
tains just the information required, clearly ex- 
plained in a practical manner. 

It contains 550 pages with 300 illustrations 
of electrical appliances, and is bound in heavy 
red leather, size 4!^x6_J^ for the pocket. 

Hawkins* Engineers* Examinations , 

This work is an important aid to engineers 
of all grades, and is undoubtedly the most help- 
ful ever issued relating to a safe and sure 
preparation for examination. It presents in a 
question and answer form the most approved 
practice in the care and management of Steam 
Boilers, Engines, Pumps, Electrical and Refriger- 
ating Machines, together with much operative 
information useful to the student. 

Hawkins* Steam Engine Catechism. 

This work is gotten up to fill a long-felt need 
for a practical book. It gives directions and 
detailed descriptions for running the various 
types of steam engines in use. 

The book also treats generously upon Ma- 
rine, Locomotive and Gas Engines, and will be 
found valuable to all users of these motive 
powers, 

Hawkins* Steam Boiler Practice. 

This instructive book on Boiler Room 
Practice is indispensable to Firemen, Engineers 
and all others wishing to perfect themselves in 
this important branch of Steam Engineering. 

Besides a full descriptive treatise on Station- 
ary, Marine and Locomotive boilers, it contains 
sixty management cautions, all necessary rules 
and specifications for boilers, including riveting, 
bracing, finding pressure, strain on bolts, etc., 
thus being a complete hand-book on the sub- 
ject. 



$2 



$2 



$2 



$2 



$2 



Hawkins* Mechanical Drawing. 

This work is arranged according to the cor- 
rect principles of the art of drawing; each 
theme being clearly illustrated to aid the student 
to ready and rapid comprehension. 

It contains 310 pages with oyer 300 illustra- 
tions, including useful diagrams and suggestions 
in drawings for practice. Handsomely bound 
in dark green cloth. Size 7x10 inches. 

Hawkins' Calculations for Engineers. vO 

The Hand Book of Calculations is a work of 
instruction and reference relating to the steam 
engine, steam boiler, etc., and has been said to 
contain every calculation, rule and table neces- U 

sary for the Engineer, Fireman and Steam ',, 

User. 

It is a complete course in Mathematics. All 
calculations are in plain arithmetical figures, so 
that they can be understood at a glance. 



Hawkins' Steam Engine Indicator. 

This work is designed for the use of erecting 
and operating engineers, superintendents and 
students of steam engineering, relating, as it 
does, to the economical use of steam. 

The work is profusely illustrated with working 
cards taken from every day use, and gives many 
plain and valuable lessons derived from the 
diagrams. 

Guarantee. 

These books we guarantee to be in every 
way as represented, and if not found 
satisfactory can be returned promptly and 
the amount paidwill be willingly refunded. 
All books shipped post paid. 

Remittances are best sent by Check, 
Post Office or Express Money Orders. 



$1 



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